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 \(\mathbb {R}^2\), then the convergence rate of a finite element solution to an exact extremal function is \(O(h^2)\) in the \(L^2\) norm, and it is O(h) in the \(H^1\) norm, where h denotes the mesh size of a triangulation of the domain.

中文翻译：

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二维凸域上Sobolev不等式的极值逼近的有限元逼近的收敛速度。

我们研究基于Somlev不等式极值函数的基于FEM的数值方案。本文的主要结果表明，如果该域在\（\ mathbb {R} ^ 2 \）中是多边形且凸的，则有限元解对精确极值函数的收敛速度为\（O（h ^ 2 } \）在\（L ^ 2 \）范数中，它是O（h）在\（H ^ 1 \）范数中，其中h表示域的三角剖分的网格大小。