Efficient High Accuracy Solutions with ${\text{GMRES}}(m)$

Previous article Next article On the Numerical Solution of Degenerate Linear and Nonlinear Elliptic Boundary Value ProblemsHarold J. KushnerHarold J. Kushnerhttps://doi.org/10.1137/0705053PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Richard S. Varga, Matrix iterative analysis, Prentice-Hall Inc., Englewood Cliffs, N.J., 1962xiii+322 MR0158502 Google Scholar[2] Ronald A. Howard, Dynamic programming and Markov processes, The Technology Press of M.I.T., Cambridge, Mass., 1960viii+136 MR0118514 0091.16001 Google Scholar[3] J. H. Eaton and , L. A. Zadeh, Optimal pursuit strategies in discrete-state probabilistic systems, Trans. ASME Ser. D. J. Basic Engrg., 84 (1962), 23–29 MR0153510 CrossrefGoogle Scholar[4] William Feller, An Introduction to Probability Theory and Its Applications. Vol. I, John Wiley & Sons Inc., New York, N.Y., 1950xii+419 MR0038583 0039.13201 Google Scholar[5] H. J. Kushner, Methods for the numerical solution of degenerate linear and non-linear elliptic boundary value problems, Tech. Rep., 67-7, Center for Dynamical Systems, Brown University, Providence, 1967 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails The Numerical Solution of Singular Perturbations of Boundary Value ProblemsFred W. DorrSIAM Journal on Numerical Analysis, Vol. 7, No. 2 | 14 July 2006AbstractPDF (2164 KB)Vacancy Transients During Impurity Diffusion in SemiconductorsPhysical Review B, Vol. 1, No. 6 | 15 March 1970 Cross Ref BIBLIOGRAPHYIterative Solution of Nonlinear Equations in Several Variables | 1 Jan 1970 Cross Ref Numerical methods for the solution of the degenerate nonlinear elliptic equations arising in optimal stochastic control theoryIEEE Transactions on Automatic Control, Vol. 13, No. 4 | 1 Aug 1968 Cross Ref Volume 5, Issue 4| 1968SIAM Journal on Numerical Analysis647-889 History Submitted:15 January 1968Published online:14 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705053Article page range:pp. 664-679ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

We first give a stabilized improved moving least squares (IMLS) approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin method is provided theoretically for both linear and nonlinear elliptic boundary value problems. Finally, numerical examples are given to verify the theoretical analysis.

Linear systems of equations with positive and symmetric matrices often occur in the numerical treatment of linear and nonlinear elliptic boundary value problems. If the CG algorithm is used to solve these equations, one is able to speed up the convergence by "preconditioning." The method of preconditioning with hierarchical basis has already been considered for the Laplace equation in two space dimensions and for linear conforming elements. In the present work this method is generalized to a large class of conforming and nonconforming elements.

Morse–Smale index theorems for elliptic boundary deformation problems

A survey on various computational techniques for nonlinear elliptic boundary value problems

Factorization of linear elliptic boundary value problems in non-cylindrical domains

In this paper, an h p hp -local discontinuous Galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type. On h p hp -quasiuniform meshes, using the Brouwer fixed point theorem, it is shown that the discrete problem has a solution, and then using Lipschitz continuity of the discrete solution map, uniqueness is also proved. A priori error estimates in broken H 1 H^1 norm and L 2 L^2 norm which are optimal in h h , suboptimal in p p are derived. These results are exactly the same as in the case of linear elliptic boundary value problems. Numerical experiments are provided to illustrate the theoretical results.

The Schwarz alternating method can be used to solve linear elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which result from solving a sequence of elliptic boundary value problems in each of the subdomains. This paper considers four Schwarz alternating methods for the N-dimensional, steady, viscous, incompressible Navier--Stokes equations, $N \leq 4$. It is shown that the Schwarz sequences converge to the true solution provided that the Reynolds number is sufficiently small.

The Galerkin finite elements method (GAFEM) is used to find the numerical solution (NS) of the couple linear elliptic boundary value problem (CLEBVP). The solution of the CLEBVP is approximated by a linear combination of piecewise linear and continuous basis functions, upon substituting this approximated solution into the CLEBVP the problem then reduces to solve a “Galerkin” linear algebraic system (GALAS). This system is solved by using the classical Cholesky method (CChM). Some illustrative examples are given; to show the high efficiency of the GAFEM to solve such for the CLEBVPs. Figures provide the NS with “results”.

Via the linking theorem, the existence of nontrivial solutions for a nonlinear elliptic Dirichlet boundary value problem with an inverse square potential is proved.

We consider estimates depending on a parameter for general linear elliptic boundary value problems, with nonhomogeneous boundary conditions, in Nikol’skij spaces. These estimates are then employed to study general linear nonautonomous parabolic systems, again with nonhomogeneous boundary conditions. Maximal regularity results are proved.

We are interested in PDE based mesh generation. The mesh is computed as the solution of a mesh PDE which is coupled to the physical PDE of interest. In [3] we proposed a stochastic domain decomposition (SDD) method to find adaptive meshes for steady state problems by solving a linear elliptic mesh generator. The SDD approach, as originally formulated in [1], relies on a numerical evaluation of the probabilistic form of the exact solution of the linear elliptic boundary value problem. Monte-Carlo simulations are used to evaluate this probabilistic form only at the sub-domain interfaces. These interface approximations can be computed independently and are then used as Dirichlet boundary conditions for the deterministic sub-domain solves. It is generally not necessary to solve the mesh PDEs with high accuracy. Only a good quality mesh, one that allows an accurate representation of the physical PDE, is required. This lower accuracy requirement makes the proposed SDD method computationally more attractive, reducing the number of Monte-Carlo simulations required.

Fast iterative solution of elliptic control problems in wavelet discretization

In this paper, we describe a method which proves by computers the existence of weak solutions for linear elliptic boundary value problems of second order. It is shown that we can constitute the computing procedures to verify the existence, uniqueness and inclusion set of a solution based on Schauder’s fixed point theorem. Using the finite element approximations for some simple Poisson’s equations and the results of error estimates, we generate iteratively a set sequence composed of functions and attempt to construct automatically the set including the exact solution. Further, the conditions of verifiability by this method are considered and some numerical examples of verification are presented.

Integral representations for the solutions to linear elliptic self adjoint boundary value problems are derived in terms of two functions which are generalisations of the single and double layer potentials used in the theory of harmonic functions. The generalised potentials are constructed in terms of a fundamental solution which is an approximation to the exact kernel of the boundary value problem in question. The representations so obtained are shown to provide a basis from which strict approximations to the solutions of boundary value problems can be developed. In particular the structure of the integral equation representing the given boundary value problem is precisely determined.