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Coefficients of the singularities for elliptic boundary value problems on domains with conical points. III: finite element methods on polygonal domains
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In this article, for homogeneous strongly elliptic operators with constant coefficients on polygonal domains, different methods are used based on the expressions of the coefficients obtained in the first two parts; the dual singular function method is also generalized.Abstract:
In the two first parts of this work [RAIRO Model. Math. Anal. Numer., 24 (1990), pp. 27e52], [RAIRO Model. Math. Anal. Numer., 24 (1990), pp. 343–367] formulas giving the coefficients arising in the singular expansion of the solutions of elliptic boundary value problems on nonsmooth domains are investigated. Now, for the case of homogeneous strongly elliptic operators with constant coefficients on polygonal domains, the solution of such problems by the finite element method is considered. In order to approximate the solution or the coefficients, different methods are used based on the expressions of the coefficients that were obtained in the first two parts; the dual singular function method is also generalized.read more
Citations
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Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients
TL;DR: It is proved that convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fréchet differentiable non-linear functional of the solution is convergence.
Journal ArticleDOI
Element-free Galerkin method: Convergence of the continuous and discontinuous shape functions
Petr Krysl,Ted Belytschko +1 more
TL;DR: In this article, the authors consider numerical solutions of second-order elliptic partial differential equations, such as Laplace's equation, or linear elasticity, in two-dimensional, non-convex domains by the element-free Galerkin method (EFG).
Journal ArticleDOI
The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges
Thomas Apel,Serge Nicaise +1 more
TL;DR: In this paper, a specific finite element strategy for solving elliptic boundary value problems in domains with corners and edges is proposed, which exhibits optimal convergence rates with decreasing mesh size, and a numerical experiment is described, showing a good agreement of the calculated approximation orders with the theoretically predicted ones.
Journal ArticleDOI
A Convergent Adaptive Method for Elliptic Eigenvalue Problems
Stefano Giani,Ivan G. Graham +1 more
TL;DR: The error analysis extends the theory of convergence of adaptive methods for linear elliptic source problems to elliptic eigenvalue problems, and in particular deals with various complications which arise essentially from the nonlinearity of the eigen value problem.
Journal ArticleDOI
Superconvergence of mixed finite element methods for optimal control problems
TL;DR: The superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods is investigated and realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique.
References
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Book
The Finite Element Method for Elliptic Problems
Philippe G. Ciarlet,J. T. Oden +1 more
TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Book
Elliptic Problems in Nonsmooth Domains
TL;DR: Second-order boundary value problems in polygons have been studied in this article for convex domains, where the second order boundary value problem can be solved in the Sobolev spaces of Holder functions.
Journal ArticleDOI
Stationary stokes and Navier-Stokes systems on two-or three-dimensional domains with corners. Part I: linearized equations
TL;DR: In this paper, the Stokes system in domains with corners is studied and conditions for the problem to be Fredholm are given, and its singular functions along with those of the nonlinear problem are studied in the second part of this paper.
Journal ArticleDOI
The post‐processing approach in the finite element method—Part 2: The calculation of stress intensity factors
Ivo Babuška,A. Miller +1 more
TL;DR: In this article, the authors describe post-processing techniques for the calculation of generalized stress intensity factors in the context of a model problem and discuss two broad classes of methods, one involving an influence function, and the other related to the well-known energy release principle of fracture mechanics.
Journal ArticleDOI
Triangular elements in the finite element method
James H. Bramble,Miloš Zlámal +1 more
TL;DR: For a plane polygonal domain a and a corresponding general triangulation, the authors define classes of functions pm(x, y) which are polynomials on each triangle and which are in Cm(Q) and also belong to the Sobolev space Wn'"'1(Q).