Research

The Landau-de Gennes model for nematic liquid crystals provides the computational challenges of a second-order elliptic boundary value problem with reduced regularity in nonconvex domains aswell as exciting topological singularities called vortices for certain Dirichlet data of non-zero winding number for larger order parameter \ell (named after Ginzburg). In two dimensions it simplifies to a Ginzburg-Landau model. The energy landscape in this non-convex minimisation problem is unexpectedly rich with many stationary points of the energy functional and the local solve faces severe difficulties. The nonconforming Crouzeix-Raviart finite elements have recently been shown to allow the computation of a guaranteed lower bound of the energy for sufficiently small meshes. We present an explicit residualbased a posteriori error estimate under the assumption that the discrete solution is sufficiently close to an isolated solution D. Our adaptive algorithm relies on a rigorous mathematical a posteriori error analysis for an asymptotic regime in the semi-linear problem and very small initial mesh-sizes that resolve the various stationary points of the energy. The emphasis is on the numerical verification of optimal convergence rates in computational benchmarks for the non-conforming Crouzeix-Raviart finite element method with lower-energy bounds. The validation of a physical model gives new insight into the energies of the two known global minimisers and four other local minimisers. The vortex localisation with adaptive mesh design is studied in a third example of winding number two. In all numerical experiments, the novel adaptive algorithm recovers optimal convergence rates.

This work establishes a pathway to reconstruct material parameters from measurements within the Landau–de Gennes model for nematic liquid crystals. We present a Bayesian approach to this inverse problem and analyse its properties using given, simulated data for benchmark problems of a planar bistable nematic device. In particular, we discuss the accuracy of the Markov chain Monte Carlo approximations, confidence intervals and the limits of identifiability.

This work considers the optimization of electrode positions in head imaging by electrical impedance tomography. The study is motivated by maximizing the sensitivity of electrode measurements to conductivity changes when monitoring the condition of a stroke patient, which justifies adopting a linearized version of the complete electrode model as the forward model. The algorithm is based on finding a (locally) A-optimal measurement configuration via gradient descent with respect to the electrode positions. The efficient computation of the needed derivatives of the complete electrode model is one of the focal points. Two algorithms are introduced and numerically tested on a three-layer head model. The first one assumes a region of interest and a Gaussian prior for the conductivity in the brain, and it can be run offline, i.e., prior to taking any measurements. The second algorithm first computes a reconstruction of the conductivity anomaly caused by the stroke with an initial electrode configuration by combining lagged diffusivity iteration with sequential linearizations, which can be interpreted to produce an approximate Gaussian probability density for the conductivity perturbation. It then resorts to the first algorithm to find new, more informative positions for the available electrodes with the constructed density as the prior.

We consider a system of second-order nonlinear elliptic partial differential equations that models the equilibrium configurations of a two-dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin (dG) finite element methods are used to approximate the solutions of this nonlinear problem with nonhomogeneous Dirichlet boundary conditions. A discrete inf–sup condition demonstrates the stability of the dG discretization of a well-posed linear problem. We then establish the existence and local uniqueness of the discrete solution of the nonlinear problem. A priori error estimates in the energy and L2 norms are derived and a best approximation property is demonstrated. Further, we prove the quadratic convergence of the Newton iterates along with complementary numerical experiments.

We study a system of semi-linear elliptic partial differential equations with a lower order cubic nonlinear term, and inhomogeneous Dirichlet boundary conditions, relevant for two-dimensional bistable liquid crystal devices, within a reduced Landau–de Gennes framework. The main results are (i) a priori error estimates for the energy norm, within the Nitsche’s and discontinuous Galerkin frameworks under milder regularity assumptions on the exact solution and (ii) a reliable and efficient a posteriori analysis for a sufficiently large penalization parameter and a sufficiently fine triangulation in both cases. Numerical examples that validate the theoretical results, are presented separately.

 This work focuses on the analysis of a free energy functional, that models a dilute suspension of magnetic nanoparticles in a two-dimensional nematic well. The first partof the article is devoted to the asymptotic analysis of global energy minimizers in the limit of vanishing elastic constant, 𝓁→0 where the re-scaled elastic constant 𝓁is inversely proportional to the domain area. The first results concern the strong H1-convergence and a 𝓁-independent 𝐻2-bound for the global minimizers on smooth bounded 2D domains, with smooth boundary and topologically trivial Dirichlet conditions. The second partfocuses on the discrete approximation of regular solutions of the corresponding non-linear system of partial differential equations with cubic non-linearity and non-homogeneous Dirichlet boundary conditions. We establish (i) the existence and local uniqueness of the discrete solutions using fixed point argument, (ii) a best approximation result in energy norm, (iii) error estimates in the energy and 𝐿2norms with 𝓁-discretization parameter dependency for the conforming finite element method. Finally, the theoretical results are complemented by numerical experiments on the discrete solution profiles, the numerical convergence rates that corroborates the theoretical estimates, followed by plots that illustrate the dependence of the discretization parameter on 𝓁.

We develop a unified framework for the a priori and a posteriori error control of different lowest-order finite element methods for approximating the regular solutions of systems of partial differential equations under a set of hypotheses. The systems involve cubic nonlinearities in lower order terms, non-homogeneous Dirichlet boundary conditions, and the results are established under minimal regularity assumptions on the exact solution. The key contributions include (i) results for existence and local uniqueness of the discrete solutions using Newton–Kantorovich theorem, (ii) a priori error estimates in the energy norm, and (iii) a posteriori error estimates that steer the adaptive refinement process. The results are applied to conforming, Nitsche, discontinuous Galerkin, and weakly over penalized symmetric interior penalty schemes for variational models of ferronematics and nematic liquid crystals. The theoretical estimates are corroborated by substantive numerical results.