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.

/pdf/further-analysis-of-multilevel-monte-carlo-methods-for-2sdskjqu12.pdf

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.

/pdf/the-finite-element-method-with-anisotropic-mesh-grading-for-2gmh6qr0e0.pdf

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

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.

/pdf/a-convergent-adaptive-method-for-elliptic-eigenvalue-27lkaloawj.pdf

A Convergent Adaptive Method for Elliptic Eigenvalue Problems

T. In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods. The state and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique. We derive L 2 superconvergence properties for the flux functions along the Gauss lines and for the scalar functions at the Gauss points via mixed projections. Moreover, global L 2 superconvergence results are obtained by virtue of an interpolation postprocessing technique. Thus, based on these superconvergence estimates, some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.

/pdf/superconvergence-of-mixed-finite-element-methods-for-optimal-yeh4oy1uxl.pdf

Superconvergence of mixed finite element methods for optimal control 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

Mbaro-Saman Lubuma

Papers

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