Abstract:This paper investigates the impact of quadrature accuracy for volume and face integrals in the finite element method using $p$-th order polynomial shape functions for elliptic problems with mixed Dirichlet and Robin boundary conditions. The optimal $p$-th order $H^1$-convergence is maintained when numerical integration with algebraic precision at least $2p-2$ for volume terms and at least $2p-1$ for face terms is adopted. For $L^2$-error, we achieve optimal $O(h^{p+1})$ convergence when using quadrature rules of precision no less than $\max\{p,2p-2\}$ for volume terms and no less than $2p-1$ for face terms. Of particular significance, we present two examples to show that the above result on $L^2$-error is sharp for the linear FEM ($p=1$). When reduced to the case of Dirichlet boundary condition, our results yield improved dependence on the given data compared to the classical results established by Ciarlet, \textit{et al}. Numerical experiments are provided to illustrate the theoretical findings and confirm the necessity of specified quadrature accuracy and data regularity.
From: Juan Han [view email]
[v1]
Mon, 15 Jun 2026 13:09:33 UTC (200 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。