The error in a method's solution is defined as the difference between the approximation and the exact analytical solution. The two sources of error in finite difference methods are round-off error, the loss of precision due to computer rounding of decimal quantities, and truncation error or discretization error, the difference between the exact solution of the original differential equation and the exact quantity assuming perfect arithmetic (that is, assuming no round-off).
To use a finite difference method to approximate the solution to a problem, one must first discretize the problem's domain. This is usually done by dividing the domain into a uniform grid (see image to the right). This means that finite-difference methods produce sets of discrete numerical approximations to the derivative, often in a "time-stepping" manner.
The SBP-SAT (summation by parts - simultaneous approximation term) method is a stable and accurate technique for discretizing and imposing boundary conditions of a well-posed partial differential equation using high order finite differences.[8][9]
The method is based on finite differences where the differentiation operators exhibit summation-by-parts properties. Typically, these operators consist of differentiation matrices with central difference stencils in the interior with carefully chosen one-sided boundary stencils designed to mimic integration-by-parts in the discrete setting. Using the SAT technique, the boundary conditions of the PDE are imposed weakly, where the boundary values are "pulled" towards the desired conditions rather than exactly fulfilled. If the tuning parameters (inherent to the SAT technique) are chosen properly, the resulting system of ODE's will exhibit similar energy behavior as the continuous PDE, i.e. the system has no non-physical energy growth. This guarantees stability if an integration scheme with a stability region that includes parts of the imaginary axis, such as the fourth order Runge-Kutta method, is used. This makes the SAT technique an attractive method of imposing boundary conditions for higher order finite difference methods, in contrast to for example the injection method, which typically will not be stable if high order differentiation operators are used.
The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain.[1]The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then approximates a solution by minimizing an associated error function via the calculus of variations.
While it is difficult to quote a date of the invention of the finite element method, the method originated from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering.[4] Its development can be traced back to the work by A. Hrennikoff[5] and R. Courant[6] in the early 1940s. Another pioneer was Ioannis Argyris. In the USSR, the introduction of the practical application of the method is usually connected with name of Leonard Oganesyan.[7] It was also independently rediscovered in China by Feng Kang in the later 1950s and early 1960s, based on the computations of dam constructions, where it was called the finite difference method based on variation principle. Although the approaches used by these pioneers are different, they share one essential characteristic: mesh discretization of a continuous domain into a set of discrete sub-domains, usually called elements.
The finite element method obtained its real impetus in the 1960s and 1970s by the developments of J. H. Argyris with co-workers at the University of Stuttgart, R. W. Clough with co-workers at UC Berkeley, O. C. Zienkiewicz with co-workers Ernest Hinton, Bruce Irons[8] and others at Swansea University, Philippe G. Ciarlet at the University of Paris 6 and Richard Gallagher with co-workers at Cornell University. Further impetus was provided in these years by available open source finite element programs. NASA sponsored the original version of NASTRAN, and UC Berkeley made the finite element program SAP IV[9] widely available. In Norway the ship classification society Det Norske Veritas (now DNV GL) developed Sesam in 1969 for use in analysis of ships.[10] A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by Strang and Fix.[11] The method has since been generalized for the numerical modeling of physical systems in a wide variety of engineering disciplines, e.g., electromagnetism, heat transfer, and fluid dynamics.[12][13]
Another consideration is the relation of the finite-dimensional space V \displaystyle V to its infinite-dimensional counterpart, in the examples above H 0 1 \displaystyle H_0^1 . A conforming element method is one in which space V \displaystyle V is a subspace of the element space for the continuous problem. The example above is such a method. If this condition is not satisfied, we obtain a nonconforming element method, an example of which is the space of piecewise linear functions over the mesh which are continuous at each edge midpoint. Since these functions are in general discontinuous along the edges, this finite-dimensional space is not a subspace of the original H 0 1 \displaystyle H_0^1 .
The extended finite element method (XFEM) is a numerical technique based on the generalized finite element method (GFEM) and the partition of unity method (PUM). It extends the classical finite element method by enriching the solution space for solutions to differential equations with discontinuous functions. Extended finite element methods enrich the approximation space so that it can naturally reproduce the challenging feature associated with the problem of interest: the discontinuity, singularity, boundary layer, etc. It was shown that for some problems, such an embedding of the problem's feature into the approximation space can significantly improve convergence rates and accuracy. Moreover, treating problems with discontinuities with XFEMs suppresses the need to mesh and re-mesh the discontinuity surfaces, thus alleviating the computational costs and projection errors associated with conventional finite element methods, at the cost of restricting the discontinuities to mesh edges.
The virtual element method (VEM), introduced by BeirÃo da Veiga et al. (2013)[18] as an extension of mimetic finite difference (MFD) methods, is a generalisation of the standard finite element method for arbitrary element geometries. This allows admission of general polygons (or polyhedra in 3D) that are highly irregular and non-convex in shape. The name virtual derives from the fact that knowledge of the local shape function basis is not required, and is in fact never explicitly calculated.
Even though I feel like this question needs some improvement, I'm going to give a short answer.We use finite difference (such as central difference) methods to approximate derivatives, which in turn usually are used to solve differential equation (approximately).
The coefficients Ci are typically generated from Taylor series expansions and can be chosen to obtain a scheme with desired characteristics such as accuracy, and in the context of partial differential equations, dispersion and dissipation. For explicit finite difference schemes such as the type above, larger stencils typically have a higher order of accuracy. For this post we use a nine-point stencil which has eighth-order accuracy. We also choose a symmetric stencil, shown in the following equation.
We employ a tiling approach in which each thread block loads a tile of data from the multidimensional grid into shared memory, so that each thread in the block can access all elements of the shared memory tile as needed. How do we choose the best tile shape and size? Some experimentation is required, but characteristics of the finite difference stencil and grid size provide direction.
A better choice of tile (and thread block) size for our problem and data size is a 64 Ã 4 tile, as depicted on the right in the figure above. This tile avoids overlap altogether when calculating the x-derivative for our one-dimensional stencil on a grid of 643 since the tile contains all points in the direction of the derivative. A minimal tile would have just one pencil, or one-dimensional array of all points in a direction. This would correspond to thread blocks of only 64 threads, however, so from an occupancy standpoint it is beneficial to use multiple pencils per tile. In our finite difference code, available for download on Github, we parameterize the number of pencils to allow experimentation. In addition to loading each value of f only once, every warp of threads loads contiguous data from global memory using this tile and therefore achieves perfectly coalesced accesses to global memory.
38c6e68cf9