On the approximate solution of elliptic, self adjoint boundary value problems

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”.

Morse–Smale index theorems for elliptic boundary deformation problems

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

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

Solution of a nonlinear boundary value problem for a second-order ordinary differential equation by the method of finite differences

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.

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.

Weighted analytic regularity in polyhedra

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.

We consider the Neumann-Dirichlet domain decomposition method for the solution of linear elliptic boundary value problems. We study the following question. Suppose that the auxiliary problems on the subdomains are not solved exactly, but only with a fixed, mesh size independent accuracy. Does the speed of convergence remain mesh size independently bounded? We show that the answer is no in general, but that mesh size independent convergence can be obtained if the accuracy requirement on the subsolvers becomes increasingly severe as the mesh size tends to zero.

In this manuscript, we study a certain classical second-order fully nonlinear coupled system with generalized nonlinear coupled boundary conditions satisfying the monotone assumptions. Our new results unify the existence criteria of certain linear and nonlinear boundary value problems (BVPs) that have been previously studied on a case-by-case basis; for example, Dirichlet and Neumann are special cases. The common feature is that the solution of each BVPs lies in a sector defined by well-ordered coupled lower and upper solutions. The tools we use are the coupled lower and upper solutions approach along with some results of fixed point theory. By means of the coupled lower and upper solutions approach, the considered BVPs are logically modified to new problems, known as modified BVPs. The solution of the modified BVPs leads to the solution of the original BVPs. In our case, we only require the Nagumo condition to get a priori bound on the derivatives of the solution function. Further, we extend the results presented in (Franco et al. in Extr. Math. 18(2):153–160, 2003; Franco et al. in Appl. Math. Comput. 153:793–802, 2004; Franco and O’Regan in Arch. Inequal. Appl. 1:423–430, 2003; Asif et al. in Bound. Value Probl. 2015:134, 2015). Finally, as an application, we consider the fully nonlinear coupled mass-spring model.

Consideration of an abstract improvement algorithm leads to the following principle, which is similar to that underlying iterative refinement: By making judicious use of relatively few high accuracy computations, high accuracy solutions can be obtained very efficiently by the algorithm. This principle is applied specifically to ${\text{GMRES}}(m)$ here; it can be similarly applied to a number of other “restarted” iterative linear methods as well. Results are given for numerical experiments in solving a discretized linear elliptic boundary value problem and in computing a step of an inexact Newton method using finite differences for a discretized nonlinear elliptic boundary value problem.

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.

There are few numerical techniques available to solve third order boundary value problems. In this paper, we present Galerkin weighted residual method for constructing the numerical solution of third order linear and nonlinear boundary value problems with two point boundary conditions. The method is formulated as a rigorous matrix form. Several numerical examples of both linear and nonlinear boundary value problems in the literature are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods in the literature. Keywords: Galerkin method, Third order linear and nonlinear BVPs, Bernstein Polynomials.

Chapter 7 Preconditioned Iterative Methods for Nonselfadjoint or Indefinite Elliptic Boundary Value Problems