We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried out in Charrier et al. (SIAM J Numer Anal, 2013), to more difficult problems, posed on non-smooth domains and with discontinuities in the coefficient. For this wider class of model problem, we prove convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Frechet differentiable non-linear functional of the solution. We further improve the performance of the multilevel estimator by introducing level dependent truncations of the Karhunen---Loeve expansion of the random coefficient. Numerical results complete the paper.

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Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

Element-free Galerkin method: Convergence of the continuous and discontinuous shape functions

This paper is concerned with a specific finite element strategy for solving elliptic boundary value problems in domains with corners and edges. First, the anisotropic singular behaviour of the solution is described. Then the finite element method with anisotropic, graded meshes and piecewise linear shape functions is investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates for functions from anisotropically weighted spaces are derived. Finally, a numerical experiment is described, that shows a good agreement of the calculated approximation orders with the theoretically predicted ones. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

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The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges

Introduction. 1. Preliminaries. 2. Integral inequalities. 3. The Laplace operator. 4. Strong solutions of the Dirichlet problem for linear equations. 5. The Dirichlet problem for elliptic linear. divergent equations in a nonsmooth domain. 6. The Dirichlet problem for semilinear equations in a conical domain. 7. Strong solutions of the Dirichlet problem for nondivergence quasilinear equations. 8. Weak solutions of the Dirichlet problem for elliptic divergence form quasilinear equations. 9. The behavior of weak solutions to the boundary value problems for elliptic quasilinear equations with triple degeneration in a neighborhood of a boundary edge. 10. Sharp estimates of solutions to the Robin. boundary value problem for elliptic non divergence second order equations in a neighborhood of the conical point. Bibliography. Notation Index. Index.

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Elliptic boundary value problems of second order in piecewise smooth domains

We prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second-order symmetric elliptic partial differential operators. The weak form is assumed to yield a bilinear form which is bounded and coercive in $H^1$. Each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines elements in which the computed eigenfunction has high oscillation. 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 eigenvalue problem. Because of this nonlinearity, the convergence result holds under the assumption that the initial finite element mesh is sufficiently fine.

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A Convergent Adaptive Method for Elliptic Eigenvalue Problems

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.

https://www.jstor.org/stable/pdfplus/2158080.pdf

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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Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques. I : Résultats généraux pour le problème de Dirichlet

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Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques. II : Quelques opérateurs particuliers

Error estimates on the coeficients obtained by the singular function method

The Neumann problem for Laplace's equation in a polygonal domain is associated with the exterior Dirichlet problem obtained by requiring the continuity of the potential through the boundary. Then the solution is the simple layer potential of the charge q on the boundary. q is the solution of a Fredholm integral equation of the second kind that we solve by the Galerkin method. The charge q has a singular part due to the corners, so the optimal order of convergence is not reached with a uniform mesh. We restore this optimal order by grading the mesh adequately near the corners. The interior Dirichlet problem is solved analogously, by expressing the solution as a double layer potential.

Maryse Bourlard

Papers

Coefficients of the singularities for elliptic boundary value problems on domains with conical points. III: finite element methods on polygonal domains

Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques. I : Résultats généraux pour le problème de Dirichlet

Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques. II : Quelques opérateurs particuliers

Error estimates on the coeficients obtained by the singular function method

An adapted Galerkin method for the resolution of Dirichlet and Neumann problems in a polygonal domain