Three-scale finite element eigenvalue discretizations

Some three-scale finite element discretization schemes are proposed and analyzed in this paper for a class of elliptic eigenvalue problems on tensor product domains. With these schemes, the solution of an eigenvalue problem on a fine grid may be reduced to the solutions of eigenvalue problems on a relatively coarse grid and some partially mesoscopic grids, together with the solutions of linear algebraic systems on a globally mesoscopic grid and several partially fine grids. It is shown theoretically and numerically that this type of discretization schemes not only significantly reduce the number of degrees of freedom but also produce very accurate approximations. AMS subject classification (2000)  65N15, 65N25, 65N30, 65N50     

Convergence of fluctuation-splitting schemes for two dimensional scalar conservation laws with a kinetic solver

Summary. The “fluctuation-splitting schemes” (FSS in short) have been introduced by Roe and Sildikover to solve advection equations on rectangular grids and then extended to triangular grids by Roe, Deconinck, Struij... For a two dimensional nonlinear scalar conservation law, we consider the case of a triangular grid and of a kinetic approach to reduce the discretization of the nonlinear equation to a linear equation and apply a particular FSS called N-scheme. We show that the resulting scheme converges strongly in in a finite volume sense. Received February 25, 1997 / Revised version received November 8, 1999 / Published online August 24, 2000     

Repeated Richardson extrapolation applied to the two-dimensional Laplace equation using triangular and square grids

The focus of this work is to verify the efficiency of the Repeated Richardson Extrapolation (RRE) to reduce the discretization error in a triangular grid and to compare the result to the one obtained for a square grid for the two-dimensional Laplace equation. Two different geometries were employed: the first one, a unitary square domain, was discretized into a square or triangular grid; and the second, a half square triangle, was discretized into a triangular grid. The methodology employed used the following conditions: the finite volume method, uniform grids, second-order accurate approximations, several variables of interest, Dirichlet boundary conditions, grids with up to 16,777,216 nodes for the square domain and up to 2097,152 nodes for the half square triangle domain, multigrid method, double precision, up to eleven Richardson extrapolations for the first domain and up to ten Richardson extrapolations for the second domain. It was verified that (1) RRE is efficient in reducing the discretization error in a triangular grid, achieving an effective order of approximately 11 for all the variables of interest for the first geometry; (2) for the same number of nodes and with or without RRE, the discretization error is smaller in a square grid than in a triangular grid; and (3) the magnitude of the numerical error reduction depends on, among other factors, the variable of interest and the domain geometry.     

A comparative Fourier analysis of discontinuous Galerkin schemes for advection–diffusion with respect to BR1, BR2, and local discontinuous Galerkin diffusion discretization

This work compares the wave propagation properties of discontinuous Galerkin (DG) schemes for advection–diffusion problems with respect to the behavior of classical discretizations of the diffusion terms, that is, two versions of the local discontinuous Galerkin (LDG) scheme as well as the BR1 and the BR2 scheme. The analysis highlights a significant difference between the two possible ways to choose the alternating LDG fluxes showing that the variant that is inconsistent with the upwind advective flux is more accurate in case of advection–diffusion discretizations. Furthermore, whereas for the BR1 scheme used within a third order DG scheme on Gauss-Legendre nodes, a higher accuracy for well-resolved problems has previously been observed in the literature, this work shows that higher accuracy of the BR1 discretization only holds for odd orders of the DG scheme. In addition, this higher accuracy is generally lost on Gauss–Legendre–Lobatto nodes.     

A numerical study of 2D integrated RBFNs incorporating Cartesian grids for solving 2D elliptic differential problems

This article reports a numerical discretization scheme, based on two‐dimensional integrated radial‐basis‐function networks (2D‐IRBFNs) and rectangular grids, for solving second‐order elliptic partial differential equations defined on 2D nonrectangular domains. Unlike finite‐difference and 1D‐IRBFN Cartesian‐grid techniques, the present discretization method is based on an approximation scheme that allows the field variable and its derivatives to be evaluated anywhere within the domain and on the boundaries, regardless of the shape of the problem domain. We discuss the following two particular strengths, which the proposed Cartesian‐grid‐based procedure possesses, namely (i) the implementation of Neumann boundary conditions on irregular boundaries and (ii) the use of high‐order integration schemes to evaluate flux integrals arising from a control‐volume discretization on irregular domains. A new preconditioning scheme is suggested to improve the 2D‐IRBFN matrix condition number. Good accuracy and high‐order convergence solutions are obtained. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

The Finite Difference Based Fast Adaptive Composite Grid Method

The fast adaptive composite grid (FAC) method is an iterative method for solving discrete boundary value problems on composite grids. McCormick introduced the method in [8] and considered the convergence behaviour for discrete problems resulting from finite volume element discretization on composite grids. In this paper we consider discrete problems resulting from finite difference discretization on composite grids. We distinguish between two obvious discretization approaches at the grid points on the interfaces between fine and coarse subgrids. The FAC method for solving such discrete problems is described. In the FAC method several intergrid transfer operators appear. We study how the convergence behaviour depends on these intergrid transfer operators. Based on theoretical insights, (quasi-)optimal intergrid transfer operators are derived. Numerical results illustrate the fast convergence of the FAC method using these intergrid transfer operators.     

Analysis of a two‐level method for anisotropic diffusion equations on aligned and nonaligned grids

This paper is on the convergence analysis for two‐grid and multigrid methods for linear systems arising from conforming linear finite element discretization of the second‐order elliptic equations with anisotropic diffusion. The multigrid algorithm with a line smoother is known to behave well when the discretization grid is aligned with the anisotropic direction; however, this is not the case with a nonaligned grid. The analysis in this paper is mainly focused on two‐level algorithms. For aligned grids, a lower bound is given for a pointwise smoother, and this bound shows a deterioration in the convergence rate, whereas for ‘maximally’ nonaligned grids (with no edges in the triangulation parallel to the direction of the anisotropy), the pointwise smoother results in a robust convergence. With a specially designed block smoother, we show that, for both aligned and nonaligned grids, the convergence is uniform with respect to the anisotropy ratio and the mesh size in the energy norm. The analysis is complemented by numerical experiments that confirm the theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.     

An implementation of a discretization method for semi-infinite programming

For linear semi-infinite programming problems a discretization method is presented. A first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as starting points for the subsequent grids and on the other hand to considerably reduce the number of constraints which have to be considered in the subsequent problems. This enables an efficient treatment of large problems with moderate storage requirements. A numerically stable Simplex-algorithm is used to solve the LP-subproblems. Numerical examples from bivariate Chebyshev approximation are presented.     

A new type of boundary treatment for wave propagation

Numerical approximation of wave propagation can be done very efficiently on uniform grids. The Yee scheme is a good example. A serious problem with uniform grids is the approximation of boundary conditions at a boundary not aligned with the grid. In this paper, boundary conditions are introduced by modifying appropriate material coefficients at a few grid points close to the embedded boundary. This procedure is applied to the Yee scheme and the resulting method is proven to be \(L^2\)-stable in one space dimension. Depending on the boundary approximation technique it is of first or second order accuracy even if the boundary is located at an arbitrary point relative to the grid. This boundary treatment is applied also to a higher order discretization resulting in a third order accurate method. All algorithms have the same staggered grid structure in the interior as well as across the boundaries for efficiency. A numerical example with the extension to two space dimensions is included.     

Highly accurate numerical solutions with repeated Richardson extrapolation for 2D laplace equation

A theoretical basis is presented for the repeated Richardson extrapolation (RRE) to reduce and estimate the discretization error of numerical solutions for heat conduction. An example application is described for the 2D Laplace equation using the finite difference method, a domain discretized with uniform grids, second-order accurate approximations, several variables of interest, Dirichlet boundary conditions, grids with up to 8,193 × 8,193 nodes, a multigrid method, single, double and quadruple precisions and up to twelve Richardson extrapolations. It was found that: (1) RRE significantly reduces the discretization error (for example, from 2.25E-07 to 3.19E-32 with nine extrapolations and a 1,025 × 1,025 grid, yielding an order of accuracy of 19.1); (2) the Richardson error estimator works for numerical results obtained with RRE; (3) a higher reduction of the discretization error with RRE is achieved by using higher calculation precision, a larger number of extrapolations, a larger number of grids and correct error orders; and (4) to obtain a given value error, much less CPU time and RAM memory are required for the solution with RRE than without it.     

Finite‐difference schemes for transport‐dominated equations using special collocation nodes

We introduce finite‐difference schemes based on a special upwind‐type collocation grid, in order to obtain approximations of the solution of linear transport‐dominated advection‐diffusion problems. The method is well suited when the diffusion parameter is very small compared to the discretization parameter. A theory is developed and many numerical experiments are shown. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Optimization of finite element grids by direct energy search

One of the possible approaches to the reduction of computational costs in finite element analysis is the selection of ‘optimal grids’, which produce the ‘best’ answers, in the sense of minimizing a discretization error measure, for a fixed level of computational effort. The grid optimization problem is studied in the case of grids of similar topology having a fixed number of degrees of freedom per node. A general formulation based on weighted-residual error measures is specialized to field problems associated with a positive-definite energy functional, the minimization of which, with respect to variable node locations, is adopted as a grid optimality criterion. The problem is then embedded in the framework of the general nonlinear programming problem, and desirable computational features of candidate search algorithms are described.     

Local and parallel finite element algorithms based on two-grid discretizations

A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. The theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for finite element solutions on general shape-regular grids. Some numerical experiments are also presented to support the theory.

     

Local and Parallel Finite Element Algorithms Based on Two-Grid Discretizations for Nonlinear Problems

In this paper, some local and parallel discretizations and adaptive finite element algorithms are proposed and analyzed for nonlinear elliptic boundary value problems in both two and three dimensions. The main technique is to use a standard finite element discretization on a coarse grid to approximate low frequencies and then to apply some linearized discretization on a fine grid to correct the resulted residual (which contains mostly high frequencies) by some local/parallel procedures. The theoretical tools for analyzing these methods are some local a priori and a posteriori error estimates for finite element solutions on general shape-regular grids that are also obtained in this paper.     

Algorithm for reduced grid generation on a sphere for a global finite-difference atmospheric model

A reduced latitude-longitude grid is a modified version of a uniform spherical grid in which the number of longitudinal grid points is not fixed but depends on latitude. A method for constructing a reduced grid for a global finite-difference semi-Lagrangian atmospheric model is discussed. The key idea behind the algorithm is to generate a one-dimensional latitude grid and then to find a reduced grid that not only has a prescribed resolution structure and an admissible cell shape distortion but also minimizes a certain functional. The functional is specified as the rms interpolation error of an analytically defined function. In this way, the interpolation error, which is a major one in finite-difference semi-Lagrangian models, is taken into account. The potential of the proposed approach is demonstrated as applied to the advection equation on a sphere, which is numerically solved with various velocity fields on constructed reduced grids.     

Option pricing with a direct adaptive sparse grid approach

We present an adaptive sparse grid algorithm for the solution of the Black–Scholes equation for option pricing, using the finite element method. Sparse grids enable us to deal with higher-dimensional problems better than full grids. In contrast to common approaches that are based on the combination technique, which combines different solutions on anisotropic coarse full grids, the direct sparse grid approach allows for local adaptive refinement. When dealing with non-smooth payoff functions, this reduces the computational effort significantly. In this paper, we introduce the spatially adaptive discretization of the Black–Scholes equation with sparse grids and describe the algorithmic structure of the numerical solver. We present several strategies for adaptive refinement, evaluate them for different dimensionalities, and demonstrate their performance showing numerical results.     

Semi‐Unstructured Grids in the Laser Simulation Program LASCAD

Automatic grid generation is very important in industrial applications. Furthermore, in several applications discretization grids are needed which provide an anisotropic refinement and an accurate approximation of the derivatives of the solution. These requirements are fulfilled by semi‐unstructured grids. In this paper, we report how semi‐unstructured grids are used in the laser simulation program LASCAD [3].     

Multilevel algorithms for finite difference discretizations on sparse grids

We propose some algorithms to solve the system of linear equations arising from the finite difference discretization on sparse grids. For this, we will use the multilevel structure of the sparse grid space or its full grid subspaces, respectively.     

Analysis of the parareal approach based on discontinuous Galerkin method for time-dependent Stokes equations

This paper analyzes a parareal approach based on discontinuous Galerkin (DG) method for the time-dependent Stokes equations. A class of primal discontinuous Galerkin methods, namely variations of interior penalty methods, are adopted for the spatial discretization in the parareal algorithm (we call it parareal DG algorithm). We study three discontinuous Galerkin methods for the time-dependent Stokes equations, and the optimal continuous in time error estimates for the velocities and pressure are derived. Based on these error estimates, the proposed parareal DG algorithm is proved to be unconditionally stable and bounded by the error of discontinuous Galerkin discretization after a finite number of iterations. Finally, some numerical experiments are conducted which confirm our theoretical results, meanwhile, the efficiency of the parareal DG algorithm can be seen through a parallel experiment.     

Higher order numerical discretizations for exterior and biharmonic type PDEs

A higher order numerical discretization technique based on Minimum Sobolev Norm (MSN) interpolation was introduced in our previous work. In this article, the discretization technique is presented as a tool to solve two hard classes of PDEs, namely, the exterior Laplace problem and the biharmonic problem. The exterior Laplace problem is compactified and the resultant near singular PDE is solved using this technique. This finite difference type method is then used to discretize and solve biharmonic type PDEs. A simple book keeping trick of using Ghost points is used to obtain a perfectly constrained discrete system. Numerical results such as discretization error, condition number estimate, and solution error are presented. For both classes of PDEs, variable coefficient examples on complicated geometries and irregular grids are considered. The method is seen to have high order of convergence in all these cases through numerical evidence. Perhaps for the first time, such a systematic higher order procedure for irregular grids and variable coefficient cases is now available. Though not discussed in the paper, the idea seems to be easily generalizable to finite element type techniques as well.