Publications

Abstract: Let $\Omega\subset\mathbb{R}^d$ and consider the magnetic Laplace operator given by $H(A)=(−i\nabla−A(x))^2$, where $A:\Omega\to\mathbb{R}^d$, subject to Dirichlet boundary conditions. This operator can, for certain vector fields $A$, have eigenfunctions $H(A)\psi=\lambda\psi$ that are highly localized in a small region of $Ω$. The main goal of this paper is to show that if $|\psi|$ assumes its maximum in $x_0\in\Omega$, then $A$ behaves `almost' like a conservative vector field in a $1/\sqrt{λ}$−neighborhood of $x_0$ in a precise sense: we expect localization in regions where $|curlA|$ is small. The result is illustrated with numerical examples.

@misc{ovall2023localization,

      title={On localization of eigenfunctions of the magnetic Laplacian}, 

      author={Jeffrey S. Ovall and Hadrian Quan and Robyn Reid and Stefan Steinerberger},

      year={2023},

      eprint={2308.15994},

      archivePrefix={arXiv},

      primaryClass={math.AP}

}

Abstract: In recent years, there has been significant interest in the development of finite element methods defined on meshes that include rather general polytopes and curvilinear polygons. In the present work, we provide tools necessary to employ multiply connected mesh cells in planar domains, i.e., cells with holes, in finite element computations. Our focus is efficient evaluation of the \(H^1\) semi-inner product and \(L^2\) inner product of implicitly defined finite element functions of the types arising in boundary element based finite element methods and virtual element methods. Such functions are defined as solutions of Poisson problems having a polynomial source term and continuous boundary data. We show that the integrals of interest can be reduced to integrals along the boundaries of mesh cells, thereby avoiding the need to perform any computations in cell interiors. The dominating cost of this reduction is solving a relatively small Nyström system to obtain a Dirichlet-to-Neumann map, as well as the solution of two more Nyström systems to obtain an “anti-Laplacian” of a harmonic function, which is used for computing the \(L^2\) inner product. Several numerical examples demonstrate the high-order accuracy of this approach. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at both https://github.com/samreynoldsmath/PuncturedFEM and the supplementary materials (PuncturedFEM\_v0\_2\_5.zip [1.75MB]).

@article{doi:10.1137/23M1569332,

author = {Ovall, Jeffrey S. and Reynolds, Samuel E.},

title = {Evaluation of Inner Products of Implicitly Defined Finite Element Functions on Multiply Connected Planar Mesh Cells},

journal = {SIAM Journal on Scientific Computing},

volume = {46},

number = {1},

pages = {A338-A359},

year = {2024},

doi = {10.1137/23M1569332},

URL = { https://doi.org/10.1137/23M1569332},

eprint = {https://doi.org/10.1137/23M1569332}

,

    abstract = { Abstract. }

}

Abstract: We introduce an approach for exploring eigenvector localization phenomena for a class of (unbounded) selfadjoint operators. More specifically, given a target region and a tolerance, the algorithm identifies candidate eigen- pairs for which the eigenvector is expected to be localized in the target region to within that tolerance. Theoretical results, together with detailed numerical illustrations of them, are provided that support our algorithm. A partial realization of the algorithm is described and tested, providing a proof of concept for the approach.

@article {MR4550318,

    AUTHOR = {Ovall, Jeffrey S. and Reid, Robyn},

     TITLE = {An algorithm for identifying eigenvectors exhibiting strong

              spatial localization},

   JOURNAL = {Math. Comp.},

  FJOURNAL = {Mathematics of Computation},

    VOLUME = {92},

      YEAR = {2023},

    NUMBER = {341},

     PAGES = {1005--1031},

      ISSN = {0025-5718,1088-6842},

   MRCLASS = {65N25 (65F15)},

  MRNUMBER = {4550318},

       DOI = {10.1090/mcom/3734},

       URL = {https://doi.org/10.1090/mcom/3734},

}

Abstract: In this paper, we show that the so-called Korn inequality holds for vector fields with a zero normal or tangential trace on a subset (of positive measure) of the boundary of Lipschitz domains. We further show that the validity of this inequality depends on the geometry of this subset of the boundary. We then consider three eigenvalue problems for the Lamé operator: we constrain the traction in the tangential direction and the normal component of the displacement, the related problem of constraining the normal component of the traction and the tangential component of the displacement, and a third eigenproblem that considers mixed boundary conditions. We show that eigenpairs for these eigenproblems exist on a broad variety of domains. Analytic solutions for some of these eigenproblems are given on simple domains.

@article {MR4489612,

    AUTHOR = {Dom\'{\i}nguez-Rivera, Sebasti\'{a}n A. and Nigam, Nilima and

              Ovall, Jeffrey S.},

     TITLE = {Korn's inequality and eigenproblems for the {L}am\'{e}

              operator},

   JOURNAL = {Comput. Methods Appl. Math.},

  FJOURNAL = {Computational Methods in Applied Mathematics},

    VOLUME = {22},

      YEAR = {2022},

    NUMBER = {4},

     PAGES = {821--837},

      ISSN = {1609-4840,1609-9389},

   MRCLASS = {65N25 (35Pxx 47A75 74B05)},

  MRNUMBER = {4489612},

       DOI = {10.1515/cmam-2021-0144},

       URL = {https://doi.org/10.1515/cmam-2021-0144},

}

Abstract: H1-conforming Galerkin methods on polygonal meshes such as VEM, BEM-FEM and Trefftz-FEM employ local finite element functions that are implicitly defined as solutions of Poisson problems having polynomial source and boundary data. Recently, such methods have been extended to allow for mesh cells that are curvilinear polygons. Such extensions present new challenges for determining suitable quadratures. We describe an approach for integrating products of these implicitly defined functions, as well as products of their gradients, that reduces integrals on cells to integrals along their boundaries. Numerical experiments illustrate the practical performance of the proposed methods.

@article {MR4359368,

    AUTHOR = {Ovall, Jeffrey S. and Reynolds, Samuel E.},

     TITLE = {Quadrature for implicitly-defined finite element functions on

              curvilinear polygons},

   JOURNAL = {Comput. Math. Appl.},

  FJOURNAL = {Computers \& Mathematics with Applications. An International

              Journal},

    VOLUME = {107},

      YEAR = {2022},

     PAGES = {1--16},

      ISSN = {0898-1221,1873-7668},

   MRCLASS = {65N30 (65D30 65N50)},

  MRNUMBER = {4359368},

       DOI = {10.1016/j.camwa.2021.12.003},

       URL = {https://doi.org/10.1016/j.camwa.2021.12.003},

}

Abstract: We propose an a posteriori error estimator for high-order p- or hp-finite element discretizations of selfadjoint linear elliptic eigenvalue problems that is appropriate for estimating the error in the approximation of an eigenvalue cluster and the corresponding invariant sub- space. The estimator is based on the computation of approximate error functions in a space that complements the one in which the approximate eigenvectors were computed. These error functions are used to construct estimates of collective measures of error, such as the Hausdorff distance between the true and approximate clusters of eigenvalues, and the subspace gap between the corresponding true and approximate invariant subspaces. Numerical experiments demonstrate the practical effectivity of the approach.

@article {MR4288306,

    AUTHOR = {Giani, Stefano and Grubi\v{s}i\'{c}, Luka and Hakula, Harri

              and Ovall, Jeffrey S.},

     TITLE = {A posteriori error estimates for elliptic eigenvalue problems

              using auxiliary subspace techniques},

   JOURNAL = {J. Sci. Comput.},

  FJOURNAL = {Journal of Scientific Computing},

    VOLUME = {88},

      YEAR = {2021},

    NUMBER = {3},

     PAGES = {Paper No. 55, 25},

      ISSN = {0885-7474,1573-7691},

   MRCLASS = {65N30 (65N15 65N25 65N50)},

  MRNUMBER = {4288306},

MRREVIEWER = {Mohammad\ Asadzadeh},

       DOI = {10.1007/s10915-021-01572-2},

       URL = {https://doi.org/10.1007/s10915-021-01572-2},

}

Abstract: We present a Trefftz-type finite element method on meshes consisting of curvilinear polygons. Local basis functions are computed using integral equation techniques that allow for the efficient and accurate evaluation of quantities needed in the formation of local stiffness matrices. To define our local finite element spaces in the presence of curved edges, we must also properly define what it means for a function defined on a curved edge to be ``polynomial" of a given degree on that edge. We consider two natural choices, before settling on the one that yields the inclusion of complete polynomial spaces in our local finite element spaces, and discuss how to work with these edge polynomial spaces in practice. An interpolation operator is introduced for the resulting finite elements, and we prove that it provides optimal order convergence for interpolation error under reasonable assumptions. We provide a description of the integral equation approach used for the examples in this paper, which was recently developed precisely with these applications in mind. A few numerical examples illustrate this optimal order convergence of the finite element solution on some families of meshes in which every element has at least one curved edge. We also demonstrate that it is possible to exploit the approximation power of locally singular functions that may exist in our finite element spaces in order to achieve optimal order convergence without the typical adaptive refinement toward singular points.

@article {MR4091173,

    AUTHOR = {Anand, Akash and Ovall, Jeffrey S. and Reynolds, Samuel E. and

              Wei\ss er, Steffen},

     TITLE = {Trefftz finite elements on curvilinear polygons},

   JOURNAL = {SIAM J. Sci. Comput.},

  FJOURNAL = {SIAM Journal on Scientific Computing},

    VOLUME = {42},

      YEAR = {2020},

    NUMBER = {2},

     PAGES = {A1289--A1316},

      ISSN = {1064-8275,1095-7197},

   MRCLASS = {65N30 (65N12 65N15)},

  MRNUMBER = {4091173},

MRREVIEWER = {Juan\ Pablo\ Borthagaray},

       DOI = {10.1137/19M1294046},

       URL = {https://doi.org/10.1137/19M1294046},

}

Abstract: We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation of an operator-valued contour integral of the resolvent. Resolvents on infinite-dimensional spaces are discretized in computable finite-dimensional spaces before the algorithm is applied. This study focuses on how such discretizations result in errors in the eigenspace approximations computed by the algorithm. The computed eigenspace is then used to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff distance between the computed and exact eigenvalue clusters are obtained in terms of the discretization parameters within an abstract framework. A realization of the proposed approach for a model second-order elliptic operator using a standard finite element discretization of the resolvent is described. Some numerical experiments are conducted to gauge the sharpness of the theoretical estimates.

@article {MR4011540,

    AUTHOR = {Gopalakrishnan, Jay and Grubi\v{s}i\'{c}, Luka and Ovall,

              Jeffrey},

     TITLE = {Spectral discretization errors in filtered subspace iteration},

   JOURNAL = {Math. Comp.},

  FJOURNAL = {Mathematics of Computation},

    VOLUME = {89},

      YEAR = {2020},

    NUMBER = {321},

     PAGES = {203--228},

      ISSN = {0025-5718,1088-6842},

   MRCLASS = {65M12 (30E20 35P15 47A75 65F15)},

  MRNUMBER = {4011540},

       DOI = {10.1090/mcom/3483},

       URL = {https://doi.org/10.1090/mcom/3483},

}

Abstract:  A filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known as `FEAST', has gained considerable attention in recent years. This work studies issues that arise when FEAST is applied to compute part of the spectrum of an unbounded partial differential operator. Specifically, when the resolvent of the partial differential operator is approximated by the discontinuous Petrov-Galerkin (DPG) method, it is shown that there is no spectral pollution. The theory also provides bounds on the discretization errors in the spectral approximations. Numerical experiments for simple operators illustrate the theory and also indicate the value of the algorithm beyond the confines of the theoretical assumptions. The utility of the algorithm is illustrated by applying it to compute guided transverse core modes of a realistic optical fiber.

@article {MR3935888,

    AUTHOR = {Gopalakrishnan, Jay and Grubi\v{s}i\'{c}, Luka and Ovall,

              Jeffrey and Parker, Benjamin},

     TITLE = {Analysis of {FEAST} spectral approximations using the {DPG}

              discretization},

   JOURNAL = {Comput. Methods Appl. Math.},

  FJOURNAL = {Computational Methods in Applied Mathematics},

    VOLUME = {19},

      YEAR = {2019},

    NUMBER = {2},

     PAGES = {251--266},

      ISSN = {1609-4840,1609-9389},

   MRCLASS = {65N25 (47A10 47A75 65N30)},

  MRNUMBER = {3935888},

       DOI = {10.1515/cmam-2019-0030},

       URL = {https://doi.org/10.1515/cmam-2019-0030},

}

Abstract: We propose a high-order integral equation based method for evaluating interior and boundary derivatives of harmonic functions that are specified by their Dirichlet data in planar domains. The tangential derivative of the given Dirichlet data is used to form a complementary Neumann problem, whose solution is a harmonic conjugate of the function whose derivatives we seek. We use a high-order Nystr\"om method to compute the Dirichlet trace of the harmonic conjugate on the domain boundary. The tangential derivative of this harmonic conjugate, effected via an FFT, is the normal derivative of the original function. Because the original and conjugate harmonic functions are the real and imaginary parts of a complex analytic function, we are able to use Cauchy’s integral formulas to compute function values and derivatives inside the domain. Several numerical experiments, on smooth domains and domains with corners, illustrate the rapid convergence and high accuracy of the proposed approach.

@article {MR3817770,

    AUTHOR = {Ovall, Jeffrey S. and Reynolds, Samuel E.},

     TITLE = {A high-order method for evaluating derivatives of harmonic

              functions in planar domains},

   JOURNAL = {SIAM J. Sci. Comput.},

  FJOURNAL = {SIAM Journal on Scientific Computing},

    VOLUME = {40},

      YEAR = {2018},

    NUMBER = {3},

     PAGES = {A1915--A1935},

      ISSN = {1064-8275,1095-7197},

   MRCLASS = {65E05 (31A10 35J05 65D25 65T50)},

  MRNUMBER = {3817770},

MRREVIEWER = {Michael\ J.\ Carley},

       DOI = {10.1137/17M1141825},

       URL = {https://doi.org/10.1137/17M1141825},

}

Abstract:  We provide an abstract framework for analyzing discretization error for eigenvalue problems discretized by discontinuous Galerkin methods such as the local discontinuous Galerkin method and symmetric interior penalty discontinuous Galerkin method. The analysis applies to clusters of eigenvalues that may include degenerate eigenvalues. We use asymptotic perturbation theory for linear operators to analyze the dependence of eigenvalues and eigenspaces on the penalty parameter. We first formulate the DG method in the framework of quadratic forms and construct a companion infinite dimensional eigenvalue problem. With the use of the companion problem, the eigenvalue/vector error is estimated as a sum of two components. The first component can be viewed as a “non-conformity” error that we argue can be neglected in practical estimates by properly choosing the penalty parameter. The second component is estimated a posteriori using auxiliary subspace techniques, and this constitutes the practical estimate.

@article {MR3717711,

    AUTHOR = {Giani, Stefano and Grubi\v{s}i\'{c}, Luka and Hakula, Harri

              and Ovall, Jeffrey S.},

     TITLE = {An a posteriori estimator of eigenvalue/eigenvector error for

              penalty-type discontinuous {G}alerkin methods},

   JOURNAL = {Appl. Math. Comput.},

  FJOURNAL = {Applied Mathematics and Computation},

    VOLUME = {319},

      YEAR = {2018},

     PAGES = {562--574},

      ISSN = {0096-3003,1873-5649},

   MRCLASS = {65N30 (65N15 65N25)},

  MRNUMBER = {3717711},

       DOI = {10.1016/j.amc.2017.07.007},

       URL = {https://doi.org/10.1016/j.amc.2017.07.007},

}

Abstract: We consider families of finite elements on polygonal meshes, that are defined implicitly on each mesh cell as solutions of local Poisson problems with polynomial data. Functions in the local space on each mesh cell are evaluated via Nyström discretizations of associated integral equations, allowing for curvilinear polygons and non-polynomial boundary data. Several experiments demonstrate the approximation quality of interpolated functions in these spaces.

@article {MR3797037,

    AUTHOR = {Anand, Akash and Ovall, Jeffrey S. and Wei\ss er, Steffen},

     TITLE = {A {N}ystr\"{o}m-based finite element method on polygonal

              elements},

   JOURNAL = {Comput. Math. Appl.},

  FJOURNAL = {Computers \& Mathematics with Applications. An International

              Journal},

    VOLUME = {75},

      YEAR = {2018},

    NUMBER = {11},

     PAGES = {3971--3986},

      ISSN = {0898-1221,1873-7668},

   MRCLASS = {65N30},

  MRNUMBER = {3797037},

       DOI = {10.1016/j.camwa.2018.03.007},

       URL = {https://doi.org/10.1016/j.camwa.2018.03.007},

}

Abstract:   Sufficient conditions are provided for establishing equivalence between

best approximation error and projection/interpolation error in

finite-dimensional vector spaces for general (semi)norms. The results

are applied to several standard finite element spaces, modes of

interpolation and (semi)norms, and a numerical study of the dependence

on polynomial degree of constants appearing in our estimates is

provided.

@article {MR3696081,

    AUTHOR = {Bank, Randolph E. and Ovall, Jeffrey S.},

     TITLE = {Some remarks on interpolation and best approximation},

   JOURNAL = {Numer. Math.},

  FJOURNAL = {Numerische Mathematik},

    VOLUME = {137},

      YEAR = {2017},

    NUMBER = {2},

     PAGES = {289--302},

      ISSN = {0029-599X,0945-3245},

   MRCLASS = {65N30 (41A05 65N15 65N50)},

  MRNUMBER = {3696081},

MRREVIEWER = {Giuseppe\ Vacca},

       DOI = {10.1007/s00211-017-0877-7},

       URL = {https://doi.org/10.1007/s00211-017-0877-7},

}

Abstract: A posteriori error estimators based on auxiliary subspace techniques for second order elliptic problems in R^d (d ≥ 2) are considered. In this approach, the solution of a global problem is utilized as the error estimator. As the continuity and coercivity of the problem trivially leads to an efficiency bound, the main focus of this paper is to derive an analogous effectivity bound and to determine the computational complexity of the auxiliary approximation problem. With a carefully chosen auxiliary subspace, we prove that the error is bounded above by the error estimate up to oscillation terms. In addition, we show that the stiffness matrix of the auxiliary problem is spectrally equivalent to its diagonal. Several numerical experiments are presented verifying the theoretical results.

@article {MR3661099,

    AUTHOR = {Hakula, Harri and Neilan, Michael and Ovall, Jeffrey S.},

     TITLE = {A posteriori estimates using auxiliary subspace techniques},

   JOURNAL = {J. Sci. Comput.},

  FJOURNAL = {Journal of Scientific Computing},

    VOLUME = {72},

      YEAR = {2017},

    NUMBER = {1},

     PAGES = {97--127},

      ISSN = {0885-7474,1573-7691},

   MRCLASS = {65N30 (65N15)},

  MRNUMBER = {3661099},

MRREVIEWER = {Nicolae\ Tarfulea},

       DOI = {10.1007/s10915-016-0352-0},

       URL = {https://doi.org/10.1007/s10915-016-0352-0},

}

Abstract:  We present new residual estimates based on Kato’s square root theorem

for spectral approximations of non-self-adjoint differential operators

of convection– diffusion–reaction type. It is not assumed that the

eigenvalue/vector approximations are obtained from any particular

numerical method, so these estimates may be applied quite broadly. Key

eigenvalue and eigenvector error results are illustrated in the

context of an hp-adaptive finite element algorithm for spectral

computations, where it is shown that the resulting a posteriori error

estimates are reliable. The efficiency of these error estimates is

also strongly suggested empirically.

@article {MR3510017,

    AUTHOR = {Giani, Stefano and Grubi\v{s}i\'{c}, Luka and Mi\polhk{e}dlar,

              Agnieszka and Ovall, Jeffrey S.},

     TITLE = {Robust error estimates for approximations of non-self-adjoint

              eigenvalue problems},

   JOURNAL = {Numer. Math.},

  FJOURNAL = {Numerische Mathematik},

    VOLUME = {133},

      YEAR = {2016},

    NUMBER = {3},

     PAGES = {471--495},

      ISSN = {0029-599X,0945-3245},

   MRCLASS = {65N30 (65N15 65N25)},

  MRNUMBER = {3510017},

MRREVIEWER = {Mohammad\ Asadzadeh},

       DOI = {10.1007/s00211-015-0752-3},

       URL = {https://doi.org/10.1007/s00211-015-0752-3},

}

Abstract: The Laplace operator is pervasive in many important

mathematical models, and fundamental results such as the mean value

theorem for harmonic functions, and the maximum principle for

superharmonic functions are well known. Less well known is how the

Laplacian and its powers appear naturally in a series expansion of the

mean value of a function on a ball or sphere. This result is proven

here using Taylor’s theorem and explicit values for integrals of

monomials on balls and spheres. This result allows for nonstandard

proofs of the mean value theorem and the maximum

principle. Connections are also made with the discrete Laplacian

arising from finite difference discretization.

@article {MR3482357,

    AUTHOR = {Ovall, Jeffrey S.},

     TITLE = {The {L}aplacian and mean and extreme values},

   JOURNAL = {Amer. Math. Monthly},

  FJOURNAL = {American Mathematical Monthly},

    VOLUME = {123},

      YEAR = {2016},

    NUMBER = {3},

     PAGES = {287--291},

      ISSN = {0002-9890,1930-0972},

   MRCLASS = {35J05 (35B05 35B50 41A58 65N06)},

  MRNUMBER = {3482357},

       DOI = {10.4169/amer.math.monthly.123.3.287},

       URL = {https://doi.org/10.4169/amer.math.monthly.123.3.287},

}

Abstract: We present an hp-adaptive continuous Galerkin (hp-CG) method for approximating eigenvalues of elliptic operators, and demonstrate its utility on a collection of benchmark problems having features seen in many important practical applications—for example, high-contrast discontinuous coefficients giving rise to eigenfunctions with reduced regularity. In this continuation of our benchmark study, we concentrate on providing reliability estimates for assessing eigenfunction/invariant subspace error. In particular, we use these estimates to justify the observed robustness of eigenvalue error estimates in the presence of repeated or clustered eigenvalues. We also indicate a means for obtaining efficiency estimates from the available efficiency estimates for the associated boundary value (source) problem. As in the first part of the paper we provide extensive numerical tests for comparison with other high-order methods and also extend the list of analyzed benchmark problems.

@article {MR3456228,

    AUTHOR = {Giani, Stefano and Grubi\v{s}i\'{c}, Luka and Ovall, Jeffrey S.},

     TITLE = {Benchmark results for testing adaptive finite element

              eigenvalue procedures part 2 (conforming eigenvector and

              eigenvalue estimates)},

   JOURNAL = {Appl. Numer. Math.},

  FJOURNAL = {Applied Numerical Mathematics. An IMACS Journal},

    VOLUME = {102},

      YEAR = {2016},

     PAGES = {1--16},

      ISSN = {0168-9274,1873-5460},

   MRCLASS = {65N30 (65N15 65N25)},

  MRNUMBER = {3456228},

MRREVIEWER = {Srinivasan\ Kesavan},

       DOI = {10.1016/j.apnum.2015.12.001},

       URL = {https://doi.org/10.1016/j.apnum.2015.12.001},

}

Abstract: We develop an a posteriori error estimate of hierarchical type for Dirichlet eigenvalue problems of the form $(-\Delta + (c/r)^2)\psi = \lambda\psi$ on bounded domains $\Omega$, where $r$ is the distance to the origin, which is assumed to be in $\Omega$. This error estimate is proven to be asymptotically identical to the eigenvalue approximation error on a family of geometrically-graded meshes. Numerical experiments demonstrate this asymptotic exactness in practice.

@article {MR3356534,

    AUTHOR = {Li, Hengguang and Ovall, Jeffrey S.},

     TITLE = {A posteriori eigenvalue error estimation for a

              {S}chr\"{o}dinger operator with inverse square potential},

   JOURNAL = {Discrete Contin. Dyn. Syst. Ser. B},

  FJOURNAL = {Discrete and Continuous Dynamical Systems. Series B. A Journal

              Bridging Mathematics and Sciences},

    VOLUME = {20},

      YEAR = {2015},

    NUMBER = {5},

     PAGES = {1377--1391},

      ISSN = {1531-3492,1553-524X},

   MRCLASS = {65N30 (35J10 35J25 65N15 65N25)},

  MRNUMBER = {3356534},

MRREVIEWER = {V.\ L.\ Makarov},

       DOI = {10.3934/dcdsb.2015.20.1377},

       URL = {https://doi.org/10.3934/dcdsb.2015.20.1377},

}

Abstract: We develop an a posteriori error estimate for mixed boundary value problems of the form $(−\Delta + V )u = f$ , where the potential $V$ may possess inverse-square singularities at finitely many points in the domain. We prove that our error estimate can be efficiently computed and is asymptotically identical to the actual error in the energy norm, on a family of geometrically graded meshes appropriate for singular solutions of such problems. Therefore, our estimate can be used for a practical stopping criterion. A variety of numerical experiments support our theoretical results. We also offer a direct convergence and effectivity comparison between the geometrically-graded meshes, which are based on a priori knowledge of possible singularities in the solution, and adaptively refined meshes driven by local error indicators associated with our a posteriori error estimate.

@article {MR3276872,

    AUTHOR = {Li, Hengguang and Ovall, Jeffrey S.},

     TITLE = {A posteriori error estimation of hierarchical type for the

              {S}chr\"{o}dinger operator with inverse square potential},

   JOURNAL = {Numer. Math.},

  FJOURNAL = {Numerische Mathematik},

    VOLUME = {128},

      YEAR = {2014},

    NUMBER = {4},

     PAGES = {707--740},

      ISSN = {0029-599X,0945-3245},

   MRCLASS = {65N30 (65N15 65N50)},

  MRNUMBER = {3276872},

MRREVIEWER = {Nicolae\ Pop},

       DOI = {10.1007/s00211-014-0628-y},

       URL = {https://doi.org/10.1007/s00211-014-0628-y},

}

Abstract: We consider a two-level block Gauss–Seidel iteration for solving systems arising from finite element discretizations employing higher-order elements. A p-hierarchical basis is used to induce this block structure. Using superconvergence results normally employed in the analysis of gradient recovery schemes, we argue that a massive reduction of the H1-error occurs in the first iterate, so that the discrete solution is adequately resolved in very few iterates—sometimes a single iteration is sufficient. Numerical experiments on uniform and adapted meshes support these claims.

@article {MR3105296,

    AUTHOR = {Le Borne, S. and Ovall, J. S.},

     TITLE = {Rapid error reduction for block {G}auss-{S}eidel based on

              {$p$}-hierarchical basis},

   JOURNAL = {Numer. Linear Algebra Appl.},

  FJOURNAL = {Numerical Linear Algebra with Applications},

    VOLUME = {20},

      YEAR = {2013},

    NUMBER = {5},

     PAGES = {743--760},

      ISSN = {1070-5325,1099-1506},

   MRCLASS = {65F10},

  MRNUMBER = {3105296},

MRREVIEWER = {Gerard\ Awanou},

       DOI = {10.1002/nla.1841},

       URL = {https://doi.org/10.1002/nla.1841},

}

Abstract: We present reliable a-posteriori errorestimates for hp-adaptive finite element approximations of semi-definite eigenvalue/eigenvector problems. Our model problems are motivated by applications in photonic crystal eigenvalue computations. We present detailed numerical experiments confirming our theory and give several benchmark results which could serve the purpose of numerical testing of other adaptive procedures.

@article {MR3054372,

    AUTHOR = {Giani, Stefano and Grubi\v{s}i\'{c}, Luka and Ovall, Jeffrey S.},

     TITLE = {Error control for {$hp$}-adaptive approximations of

              semi-definite eigenvalue problems},

   JOURNAL = {Computing},

  FJOURNAL = {Computing. Archives for Scientific Computing},

    VOLUME = {95},

      YEAR = {2013},

    NUMBER = {1},

     PAGES = {S235--S257},

      ISSN = {0010-485X,1436-5057},

   MRCLASS = {65N25 (65N15 65N30)},

  MRNUMBER = {3054372},

MRREVIEWER = {Necdet\ Bildik},

       DOI = {10.1007/s00607-012-0260-6},

       URL = {https://doi.org/10.1007/s00607-012-0260-6},

}

Abstract: We present a general framework for the a posteriori estimation and enhancement of error in eigenvalue/eigenvector computations for symmetric and elliptic eigenvalue problems, and provide detailed analysis of a specific and important example within this framework—finite element methods with continuous, affine elements. A distinguishing feature of the proposed approach is that it provides provably efficient and reliable error estimation under very realistic assumptions, not only for single, simple eigenvalues, but also for clusters which may contain degenerate eigenvalues. We reduce the study of the eigenvalue/eigenvector error estimators to the study of associated boundary value problems, and make use of the wealth of knowledge available for such problems. Our choice of a posteriori error estimator, computed using hierarchical bases, very naturally offers a means not only for estimating error in eigenvalue/eigenvector computations, but also cheaply accelerating the convergence of these computations—sometimes with convergence rates which are nearly twice that of the unaccelerated approximations.

@article {MR3018645,

    AUTHOR = {Bank, Randolph E. and Grubi\v{s}i\'{c}, Luka and Ovall, Jeffrey S.},

     TITLE = {A framework for robust eigenvalue and eigenvector error

              estimation and {R}itz value convergence enhancement},

   JOURNAL = {Appl. Numer. Math.},

  FJOURNAL = {Applied Numerical Mathematics. An IMACS Journal},

    VOLUME = {66},

      YEAR = {2013},

     PAGES = {1--29},

      ISSN = {0168-9274,1873-5460},

   MRCLASS = {65F15},

  MRNUMBER = {3018645},

       DOI = {10.1016/j.apnum.2012.11.004},

       URL = {https://doi.org/10.1016/j.apnum.2012.11.004},

}

Abstract: We present several well-posed, well-conditioned direct and indirect integral equation formulations for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions in domains with corners. We focus mainly on Direct Regularized Combined Field Integral Equation (DCFIE-R) formulations whose name reflects that (1) they consist of combinations of direct boundary integral equations of the second-kind and first-kind integral equations which are preconditioned on the left by coercive boundary single-layer operators, and (2) their unknowns are physical quantities, i.e., the total field on the boundary of the scatterer. The DCFIE-R equations are shown to be uniquely solvable in appropriate function spaces under certain assumptions on the coupling parameter. Using Calderón’s identities and the fact that the unknowns are bounded in the neigh- borhood of the corners, the integral operators that enter the DCFIE-R formulations are recast in a form that involves in- tegral operators that are expressed by convergent integrals only. The polynomially-graded mesh quadrature introduced by Kress [30] enables the high-order resolution of the weak singularities of the kernels of the integral operators and the singularities in the derivatives of the unknowns in the vicinity of the corners. This approach is shown to lead to an efficient, high-order Nyström method capable of producing solutions of sound-hard scattering problems in domains with corners which require small numbers of Krylov subspace iterations throughout the frequency spectrum. We present a variety of numerical results that support our claims.

@article {MR2993110,

    AUTHOR = {Anand, Akash and Ovall, Jeffrey S. and Turc, Catalin},

     TITLE = {Well-conditioned boundary integral equations for

              two-dimensional sound-hard scattering problems in domains with

              corners},

   JOURNAL = {J. Integral Equations Appl.},

  FJOURNAL = {Journal of Integral Equations and Applications},

    VOLUME = {24},

      YEAR = {2012},

    NUMBER = {3},

     PAGES = {321--358},

      ISSN = {0897-3962,1938-2626},

   MRCLASS = {35J05 (35P25 44A15 65R20)},

  MRNUMBER = {2993110},

MRREVIEWER = {Michael\ A.\ Perelmuter},

       DOI = {10.1216/JIE-2012-24-3-321},

       URL = {https://doi.org/10.1216/JIE-2012-24-3-321},

}

Abstract: A discontinuous Galerkin method, with hp-adaptivity based on the approximate solution of appropriate dual problems, is employed for highly-accurate eigenvalue computations on a collection of benchmark examples. After demonstrating the effectivity of our computed error estimates on a few well-studied examples, we present results for several examples in which the coefficients of the partial-differential operators are discontinuous. The problems considered here are put forward as benchmarks upon which other adaptive methods for computing eigenvalues may be tested, with results compared to our own.

@article {MR2863096,

    AUTHOR = {Giani, Stefano and Grubi\v{s}i\'{c}, Luka and Ovall, Jeffrey

              S.},

     TITLE = {Benchmark results for testing adaptive finite element

              eigenvalue procedures},

   JOURNAL = {Appl. Numer. Math.},

  FJOURNAL = {Applied Numerical Mathematics. An IMACS Journal},

    VOLUME = {62},

      YEAR = {2012},

    NUMBER = {2},

     PAGES = {121--140},

      ISSN = {0168-9274,1873-5460},

   MRCLASS = {65N30 (65N15 65N25)},

  MRNUMBER = {2863096},

MRREVIEWER = {Christian\ Wieners},

       DOI = {10.1016/j.apnum.2011.10.007},

       URL = {https://doi.org/10.1016/j.apnum.2011.10.007},

}

Abstract: We present a parallel goal-oriented adaptive finite element method that can be used to rapidly compute highly accurate solutions for 2.5-D controlled-source electromagnetic (CSEM) and 2-D magnetotelluric (MT) modelling problems. We employ unstructured triangular grids that permit efficient discretization of complex modelling domains such as those containing topography, dipping layers and multiple scale structures. Iterative mesh refinement is guided by a goal-oriented error estimator that considers the relative error in the strike aligned fields and their spatial gradients, resulting in a more efficient mesh refinement than possible with a previous approach based on the absolute errors. Reliable error estimation is accomplished by a dual weighted residual method that is carried out via hierarchical basis computations. Our algorithm is parallelized over frequencies, wavenumbers, transmitters and receivers, where adaptive refinement is performed in parallel on subsets of these parameters. Mesh sharing allows an adapted mesh generated for a particular frequency and wavenumber to be shared with nearby frequencies and wavenumbers, thereby efficiently reducing the parallel load of the adaptive refinement calculations. We demonstrate the performance of our algorithm on a large cluster computer through scaling tests for a complex model that includes strong seafloor topography variations and multiple thin stacked hydrocarbon reservoirs. In tests using up to 800 processors and a realistic suite of CSEM data parameters, our algorithm obtained run-times as short as a few seconds to tens of seconds.

@Article{KeyOvallGJI,

  author    = {Key, Kerry and Ovall, Jeffrey},

  title     = {A parallel goal-oriented adaptive finite element method for 2.5-D electromagnetic modelling},

  journal   = {Geophysical Journal International},

  year      = {2011},

  volume    = {186},

  number    = {1},

  pages     = {137--154},

  issn      = {1365-246X},

  doi       = {10.1111/j.1365-246X.2011.05025.x},

  url       = { https://doi.org/10.1111/j.1365-246X.2011.05025.x},

}

Abstract: In this article, we develop and analyze a hierarchical-type error estimator for a general class of second-order linear elliptic boundary value problems in bounded three-dimensional domains. This type of indicator automatically satisfies a global lower bound inequality, thereby giving efficiency, without regularity assumptions beyond those giving well-posedness of the continuous and discrete problems. The main focus of the paper is then to establish the reverse reliability result: a global upper bound on the error in terms of the error estimate (plus an oscillation term), again without additional regularity assumptions. The proof of this inequality depends on a clever choice of the space in which the error indicator lies and a moment-capturing quasi-interpolation result. We finish the article with a series of numerical experiments to illustrate the behavior predicted by the theoretical results.

@article {MR2772278,

    AUTHOR = {Holst, Michael and Ovall, Jeffrey S. and Szypowski, Ryan},

     TITLE = {An efficient, reliable and robust error estimator for elliptic

              problems in {$\Bbb R^3$}},

   JOURNAL = {Appl. Numer. Math.},

  FJOURNAL = {Applied Numerical Mathematics. An IMACS Journal},

    VOLUME = {61},

      YEAR = {2011},

    NUMBER = {5},

     PAGES = {675--695},

      ISSN = {0168-9274,1873-5460},

   MRCLASS = {65N15 (65N30)},

  MRNUMBER = {2772278},

MRREVIEWER = {Jeffrey\ M.\ Connors},

       DOI = {10.1016/j.apnum.2011.01.002},

       URL = {https://doi.org/10.1016/j.apnum.2011.01.002},

}

Abstract: We present a new algorithm, based on integral equation formulations, for the solution of constant-coefficient elliptic partial differential equations (PDE) in closed two-dimensional domains with non-smooth boundaries; we focus on cases in which the integral-equation solutions as well as physically meaningful quantities (such as, stresses, electric/magnetic fields, etc.) tend to infinity at singular boundary points (corners). While, for simplicity, we restrict our discussion to integral equations associated with the Neumann problem for the Laplace equation, the proposed methodology applies to integral equations arising from other types of PDEs, including the Helmholtz, Maxwell, and linear elasticity equations. Our numerical results demonstrate excellent convergence as discretizations are refined, even around singular points at which solutions tend to infinity. We demonstrate the efficacy of this algorithm through applications to solution of Neumann problems for the Laplace operator over a variety of domains—including domains containing extremely sharp concave and convex corners, with angles as small as $\pi/100$ and as large as $199\pi/100$.

@article {MR2506259,

    AUTHOR = {Bruno, Oscar P. and Ovall, Jeffrey S. and Turc, Catalin},

     TITLE = {A high-order integral algorithm for highly singular {PDE}

              solutions in {L}ipschitz domains},

   JOURNAL = {Computing},

  FJOURNAL = {Computing. Archives for Scientific Computing},

    VOLUME = {84},

      YEAR = {2009},

    NUMBER = {3-4},

     PAGES = {149--181},

      ISSN = {0010-485X,1436-5057},

   MRCLASS = {65N99 (65R20)},

  MRNUMBER = {2506259},

MRREVIEWER = {Michael\ J.\ Carley},

       DOI = {10.1007/s00607-009-0031-1},

       URL = {https://doi.org/10.1007/s00607-009-0031-1},

}

Abstract: We consider a large class of residuum based a posteriori eigenvalue/eigenvector estimates and present an abstract framework for proving their asymptotic exactness. Equivalence of the estimator and the error is also established. To demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for Laplace eigenvalue problems in planar polygonal regions. To this end we develop new error analysis for the Galerkin approximation which avoids the use of the strengthened Cauchy-Schwarz inequality and the saturation assumption, and gives reasonable and explicitly computable upper bounds on the discretization error. A brief discussion is also given concerning the design of estimators which are in the same spirit, but are based on different a posteriori techniques—notably, those of gradient recovery type.

@article {MR2476558,

    AUTHOR = {Grubi\v{s}i\'{c}, Luka and Ovall, Jeffrey S.},

     TITLE = {On estimators for eigenvalue/eigenvector approximations},

   JOURNAL = {Math. Comp.},

  FJOURNAL = {Mathematics of Computation},

    VOLUME = {78},

      YEAR = {2009},

    NUMBER = {266},

     PAGES = {739--770},

      ISSN = {0025-5718,1088-6842},

   MRCLASS = {65N25 (65N30)},

  MRNUMBER = {2476558},

MRREVIEWER = {Lucia\ Gastaldi},

       DOI = {10.1090/S0025-5718-08-02181-9},

       URL = {https://doi.org/10.1090/S0025-5718-08-02181-9},

}

Abstract: In this paper, we investigate the effects of pollution error on the performance of the parallel adaptive finite element technique proposed by Bank and Holst in 2000 [R. E. Bank and M. Holst, SIAM J. Sci. Comput., 22 (2000), pp. 1411–1443]. In particular, we consider whether the performance of the algorithm as it was originally proposed can be improved through the use of certain dual functions which give an indication of the global influences on subdomain errors.

@article {MR2341799,

    AUTHOR = {Bank, Randolph E. and Ovall, Jeffrey S.},

     TITLE = {Dual functions for a parallel adaptive method},

   JOURNAL = {SIAM J. Sci. Comput.},

  FJOURNAL = {SIAM Journal on Scientific Computing},

    VOLUME = {29},

      YEAR = {2007},

    NUMBER = {4},

     PAGES = {1511--1524},

      ISSN = {1064-8275,1095-7197},

   MRCLASS = {65N30},

  MRNUMBER = {2341799},

MRREVIEWER = {Rolf\ Stenberg},

       DOI = {10.1137/060668304},

       URL = {https://doi.org/10.1137/060668304},

}

Abstract: In this paper, we investigate the effectiveness of hierarchical matrix techniques when used as the linear solver in a certain domain decomposition algorithm. In particular, we provide a direct performance com- parison between an algebraic multigrid solver and a hierarchical matrix solver which is based on nested dissection clustering within the software package PLTMG.

@article {MR2336232,

    AUTHOR = {Ovall, J. S.},

     TITLE = {Hierarchical matrix techniques for a domain decomposition

              algorithm},

   JOURNAL = {Computing},

  FJOURNAL = {Computing. Archives for Scientific Computing},

    VOLUME = {80},

      YEAR = {2007},

    NUMBER = {4},

     PAGES = {287--297},

      ISSN = {0010-485X,1436-5057},

   MRCLASS = {65N55},

  MRNUMBER = {2336232},

MRREVIEWER = {Luca\ F.\ Pavarino},

       DOI = {10.1007/s00607-007-0235-1},

       URL = {https://doi.org/10.1007/s00607-007-0235-1},

}

Abstract: An approximate error function for the discretization error

on a given mesh is obtained by projecting (via the energy inner

product) the functional residual onto the space of continuous,

piecewise quadratic functions which vanish on the vertices of the

mesh. Conditions are given under which one can expect this

hierarchical basis error estimator to give efficient and reliable

function recovery, asymptotically exact gradient recovery, and

convergent Hessian recovery in the square norms. One does not find

similar function recovery results in the literature. The analysis

given here is based on a certain superconvergence result which has

been used elsewhere in the analysis of gradient recovery

methods. Numerical experiments are provided which demonstrate the

effectivity of the approximate error function in practice.

@article {MR2318802,

    AUTHOR = {Ovall, Jeffrey S.},

     TITLE = {Function, gradient, and {H}essian recovery using quadratic

              edge-bump functions},

   JOURNAL = {SIAM J. Numer. Anal.},

  FJOURNAL = {SIAM Journal on Numerical Analysis},

    VOLUME = {45},

      YEAR = {2007},

    NUMBER = {3},

     PAGES = {1064--1080},

      ISSN = {0036-1429,1095-7170},

   MRCLASS = {65N30 (65N15)},

  MRNUMBER = {2318802},

MRREVIEWER = {Kenneth\ Wright},

       DOI = {10.1137/060648908},

       URL = {https://doi.org/10.1137/060648908},

}

Abstract: The use of dual/adjoint problems for approximating functionals of solutions of PDEs with great accuracy or to merely drive a goal-oriented adaptive refinement scheme has become well-accepted, and it continues to be an active area of research. The traditional approach involves dual residual weighting (DRW). In this work we present two new functional error estimators and give conditions under which we can expect them to be asymptotically exact. The first is of DRW type and is derived for meshes in which most triangles satisfy an $\mathcal{O}(h^2)$-approximate parallelogram property. The second functional estimator involves dual error estimate weighting (DEW) using any superconvergent gradient recovery technique for the primal and dual solutions. Several experiments are done which demonstrate the asymptotic exactness of a DEW estimator which uses a gradient recovery scheme proposed by Bank and Xu, and the effectiveness of refinement done with respect to the corresponding local error indicators.

@article {MR2207272,

    AUTHOR = {Ovall, Jeffrey S.},

     TITLE = {Asymptotically exact functional error estimators based on

              superconvergent gradient recovery},

   JOURNAL = {Numer. Math.},

  FJOURNAL = {Numerische Mathematik},

    VOLUME = {102},

      YEAR = {2006},

    NUMBER = {3},

     PAGES = {543--558},

      ISSN = {0029-599X,0945-3245},

   MRCLASS = {65N15 (65N30 65N50)},

  MRNUMBER = {2207272},

MRREVIEWER = {Martin\ Vohral\'{\i}k},

       DOI = {10.1007/s00211-005-0655-9},

       URL = {https://doi.org/10.1007/s00211-005-0655-9},

}

Abstract:  Let $S = \{x_1, x_2, \ldots , x_n \} $ be a set of distinct positive integers such that $\gcd(x_i, x_j ) \in S$ for $1 ≤ i, j ≤ n$. Such a set is called GCD-closed. In 1875/1876, H.J.S. Smith showed that, if the set S is “factor-closed”, then the determinant of the matrix $e_{ij} = \gcd(x_i,x_j)$ is $det(E) = \prod_{m=1}^n \phi(x_m)$, where $\phi$ denotes Euler’s Phi-function. Since the early 1990’s there has been a rebirth of interest in matrices defined in terms of arithmetic functions defined on S. In 1992, Bourque and Ligh conjectured that the matrix $f_{ij} = \lcm(x_i , x_j )$ is nonsingular. Several authors have shown that, although the conjecture holds for $n \leq 7$, it need not hold in general. At present there are no known necessary conditions for F to be nonsingular, but many have offered sufficient conditions.

In this note, a simple algorithm is offered for computing the LDLT -Factorization of any matrix $b_{ij} = f(\gcd(x_i,x_j))$, where $f : S \to \mathbb{C}$. This factorization gives us an easy way of answering the question of singularity, computing its determinant, and determining its inertia (the number of positive negative and zero eigenvalues). Using this factorization, it is argued that E is positive definite regardless of whether or not S is GCD-closed (a known result), and that F is indefinite for $n \geq 2$. Also revisited are some of the known sufficient conditions for the invertibility of F , which are justified in the present framework, and then a few new sufficient conditions are offered. Similar statements are made for the reciprocal matrices $gij = \gcd(x_i,x_j)/\lcm(x_i,x_j)$ and $h_{ij} = \lcm(x_i,x_j)/\gcd(x_i,x_j)$.

@article {MR2033477,

    AUTHOR = {Ovall, Jeffrey S.},

     TITLE = {An analysis of {GCD} and {LCM} matrices via the

              {$LDL^T$}-factorization},

   JOURNAL = {Electron. J. Linear Algebra},

  FJOURNAL = {Electronic Journal of Linear Algebra},

    VOLUME = {11},

      YEAR = {2004},

     PAGES = {51--58},

      ISSN = {1081-3810},

   MRCLASS = {15A23 (06A07 06A12 15A36)},

  MRNUMBER = {2033477},

MRREVIEWER = {Bo\ Zhou},

       DOI = {10.13001/1081-3810.1121},

       URL = {https://doi.org/10.13001/1081-3810.1121},

}