Guaranteed a posteriori local error estimation for finite element solutions of boundary value problems

TL;DR

This paper introduces a guaranteed a posteriori local error estimator for finite element solutions of Poisson's equation, providing explicit bounds and improved efficiency, applicable even without $H^2$-regularity.

Contribution

It proposes a new sharp, explicit local error estimation method based on the hypercircle approach, effective for non-uniform meshes and less regular solutions.

Findings

Provides explicit error bounds for subdomains

Demonstrates efficiency on convex and non-convex domains

Applicable to problems without $H^2$-regularity

Abstract

This paper considers the finite element solution of the boundary value problem of Poisson's equation and proposes a guaranteed em a posteriori local error estimation based on the hypercircle method. Compared to the existing literature on qualitative error estimation, the proposed error estimation provides an explicit and sharp bound for the approximation error in the subdomain of interest, and its efficiency can be enhanced by further utilizing a non-uniform mesh. Such a result is applicable to problems without $H^2$-regularity, since it only utilizes the first order derivative of the solution. The efficiency of the proposed method is demonstrated by numerical experiments for both convex and non-convex 2D domains with uniform or non-uniform meshes.