Explicit Finite Element Error Estimates for Nonhomogeneous Neumann problems

TL;DR

This paper derives explicit a priori error estimates for finite element solutions to nonhomogeneous Neumann problems, providing a new bound on the trace theorem constant and demonstrating convergence through numerical examples.

Contribution

It introduces an explicit error estimate for FEM in Neumann problems, including a novel bound on the trace theorem constant.

Findings

Error estimate with convergence rate of 0.5

Explicit upper bound for trace theorem constant

Numerical validation of the error estimate

Abstract

The paper develops an explicit a priori error estimate for finite element solution to nonhomogeneous Neumann problems. For this purpose, the hypercircle equation over finite element spaces is constructed and the explicit upper bound of the constant in the trace theorem is given. Numerical examples are shown in the final section, which implies the proposed error {estimate} has the convergence rate as $0.5$.