Convergence rate of the finite element approximation for extremizers of Sobolev inequalities on 2D convex domains

초록

We investigate a FEM-based numerical scheme approximating extremal functions of the Sobolev inequalities. The main result of this paper shows that if the domain is polygonal and convex in R2, then the convergence rate of a finite element solution to an exact extremal function is O(h2) in the L2 norm, and it is O(h) in the H1 norm, where h denotes the mesh size of a triangulation of the domain.

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