The inf-sup condition and error estimates of the Nitsche method for evolutionary diffusion-advection-reaction equations

TL;DR

This paper analyzes the stability and convergence of the Nitsche method for evolutionary diffusion-advection-reaction equations, establishing inf-sup conditions and optimal error estimates for both semi-discrete and fully discrete schemes.

Contribution

It provides a variational analysis demonstrating the inf-sup condition and error estimates for the Nitsche method applied to evolutionary PDEs, including finite element and Isogeometric Analysis.

Findings

The scheme satisfies the inf-sup condition and Galerkin orthogonality.

Optimal order error estimates are proven under regularity assumptions.

Numerical examples confirm theoretical results.

Abstract

The Nitsche method is a method of "weak imposition" of the inhomogeneous Dirichlet boundary conditions for partial differential equations. This paper explains stability and convergence study of the Nitsche method applied to evolutionary diffusion-advection-reaction equations. We mainly discuss a general space semidiscrete scheme including not only the standard finite element method but also Isogeometric Analysis. Our method of analysis is a variational one that is a popular method for studying elliptic problems. The variational method enables us to obtain the best approximation property directly. Actually, results show that the scheme satisfies the inf-sup condition and Galerkin orthogonality. Consequently, the optimal order error estimates in some appropriate norms are proven under some regularity assumptions on the exact solution. We also consider a fully discretized scheme using the backward Euler method. Numerical example demonstrate the validity of those theoretical results.