On the numerical analysis of nonlinear twofold saddle point problems

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

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

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.

In this paper, we deal with the application of Half-Sweep Accelerated Over Relaxation (HSAOR) method with nonlocal discretization scheme for solving two-dimensional nonlinear elliptic boundary value problems. To do this, we propose a new nonlocal arithmetic mean scheme namely the four-point rotated nonlocal arithmetic mean scheme being imposed into any nonlinear term in the proposed problems. By using the second order finite difference scheme, the half-sweep nonlinear approximation equation has been derived. Then, the nonlocal discretization scheme is applied to transform the system of nonlinear approximation equations into the corresponding system of linear equations. Throughout numerical results, it can be pointed out that the proposed HSAOR method was superior in terms of number of iterations, execution time and maximum error compared to Full-Sweep Successive Over-relaxation (FSSOR) and Half-Sweep Successive Over Relaxation (HSSOR).

Existence theorems for nonlinear noncoercive operator equations and nonlinear elliptic boundary value problems

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

The Hopf’s maximum principles are utilized to obtain maximum principles for functions defined on solutions of nonlinear elliptic equations in divergence form (g(u)u, i ), i +f(x, u, q) = 0 (q = |∇u|2), subject to Dirichlet, Neumann or Robin boundary conditions. The principles derived may be used to deduce bounds on the gradient q.

The quasilinearization method is developed for strong solutions of semilinear and nonlinear elliptic boundary-value problems. We obtain two monotone, Lp-convergent sequences of approximate solutions. The order of convergence is two. The tools are some results on the abstract quasilinearization method and from weakly–near operators theory.

We consider symmetry properties of solutions to nonlinear elliptic boundary value problems defined on bounded symmetric domains of \({\mathbb R^n}\) . The solutions take values in ordered Banach spaces E, e.g. \({E=\mathbb R^N}\) ordered by a suitable cone. The nonlinearity is supposed to be quasimonotone increasing. By considering cones that are different from the standard cone of componentwise nonnegative elements we can prove symmetry of solutions to nonlinear elliptic systems which are not covered by previous results. We use the method of moving planes suitably adapted to cover the case of solutions of nonlinear elliptic problems with values in ordered Banach spaces.

Optimal summability of solutions to nonlinear elliptic problems

We present a derivation and implementation of asymptotic boundary conditions to be imposed on "artificial" boundaries for nonlinear elliptic boundary value problems on semi-infinite "cylindrical" domains. A general theory developed by the authors in [11] is applied to establish the existence of exact boundary conditions and then to obtain useful approximations to them. The derivation is based on the Laplace transform solution of the linearized problem at infinity. We discuss the incorporation of the asymptotic boundary conditions into a finite-difference scheme and present the results of numerical experiments on the solution of the Bratu problem in a two-dimensional stepped channel. We also touch on certain problems concerning the existence of solutions of this problem on infinite domains and conjecture on the behavior of the critical parameter value with respect to changes in the domain. Some numerical evidence supporting the conjecture is given.

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.

We report a new 9-point compact discretization of order two in y- and order four in x-directions, based on cubic spline approximation, for the solution of two-dimensional nonlinear elliptic partial differential equations of the form $$ A\left( {x,y} \right)\frac{{\partial^{2} u}}{{\partial x^{2} }} + B\left( {x,y} \right)\frac{{\partial^{2} u}}{{\partial y^{2} }} = f\left( {x,y,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}} \right),\left( {x,y} \right) \in\Omega $$ defined in the domain \( \Omega = \left\{ {\left( {x,y} \right):0 0 \) and \( B\left( {x,y} \right) > 0 \) in \( \Omega \). The corresponding Dirichlet boundary conditions are prescribed by $$ u\left( {x,y} \right) = \psi \left( {x,y} \right),\quad \left( {x,y} \right) \in \partial\Omega $$ The main spline relations are presented and incorporated into solution procedures for elliptic partial differential equations. Available numerical methods based on cubic spline approximations for the numerical solution of nonlinear elliptic equations are of second-order accurate. Although 9-point finite difference approximations of order four accurate for the solution of nonlinear elliptic differential equations are discussed in the past, but these methods require five evaluations of the function f. In this piece of work, using the same number of grid points and three evaluations of the function f, we have derived a new stable cubic spline method of order 2 in y- and order 4 in x-directions for the solution of nonlinear elliptic equation. However, for a fixed parameter (Δy/Δx 2), the proposed method behaves like a fourth order method. The accuracy of the proposed method is exhibited from the computed results. The proposed method is applicable to Poisson’s equation and two-dimensional Navier-Stokes’ equations of motion in polar coordinates, which is main highlight of the work. The convergence analysis of the proposed cubic spline approximation for the nonlinear elliptic equation is discussed and we have shown under appropriate conditions the proposed method converges. Some physical examples and their numerical results are provided to justify the advantages of the proposed method.

New extendable LU sparse factorization procedures are presented for the solution of non-linear elliptic difference equations. The derived iterative methods are shown to be both competitive and computationally efficient in comparison with existing schemes. Application of the methods on non-linear elliptic boundary value problems both in two and three space dimensions are discussed and numerical results are given.

High accuracy variable mesh method for nonlinear two-point boundary value problems in divergence form