Estimate of the Convergence Rate of Finite Element Solutions to Elliptic Equations of Second Order with Discontinuous Coefficients

TL;DR

This paper derives an asymptotically optimal error estimate for finite element solutions to second-order elliptic equations with discontinuous coefficients, highlighting the convergence rate with respect to mesh size.

Contribution

It provides the first sharp error estimate for finite element solutions to elliptic problems with discontinuous coefficients, including the effect of coefficient discontinuities.

Findings

Error estimate $ orm{u-u_k}_{1, au} o 0$ as mesh refines.

Convergence rate involves a logarithmic factor $| ext{ln} h|^{1/2}$.

Estimate is optimal in the asymptotic sense.

Abstract

In this paper, we consider elliptic boundary value problems with discontinuous coefficients and obtain the asymptotic optimal error estimate $\|u-u_k\|_{1,\Omega}\leqslant Ch|\ln h|^{1/2}\|u\|_{2,\Omega_1+\Omega_2}$ for triangle linear elements.