Graded mesh approximation in weighted Sobolev spaces and elliptic equations in 2D

James H. Adler (1) , Victor Nistor (2)

1 Department of Mathematics, Tufts University
2 IECL - Institut Élie Cartan de Lorraine

We study the approximation properties of some general finite-element spaces constructed using improved graded meshes. In our results, either the approximating function or the function to be approximated (or both) are in a weighted Sobolev space. We consider also the $L^p$-version of these spaces. The finite-element spaces that we define are obtained from {\em conformally invariant} families of finite elements (no affine invariance is used), stressing the use of elements that lead to higher regularity finite-element spaces. We prove that for a suitable grading of the meshes, one obtains the usual optimal approximation results. We provide a construction of these spaces that does not lead to long, ``skinny'' triangles. Our results are then used to obtain $L^2$ error estimates and $h^m$-quasi-optimal rates of convergence for the FEM approximation of solutions of strongly elliptic interface/boundary value problems.

Domaines: Mathématiques [math]; Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

James H. Adler, Victor Nistor. Graded mesh approximation in weighted Sobolev spaces and elliptic equations in 2D. Mathematics of Computation, 2015, 84 (295), pp.2191-2220. ⟨10.1090/S0025-5718-2015-02934-2⟩. ⟨hal-01279253⟩