The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach spaces

TL;DR

This paper introduces a minimal-residual method in discrete dual norms for solving advection-reaction equations in Banach spaces, enabling accurate approximation of irregular solutions and addressing challenges like Gibbs phenomena.

Contribution

It develops a novel minimal-residual approach in Banach spaces, analyzing its stability, well-posedness, and quasi-optimality, with practical numerical demonstrations.

Findings

Method guarantees discrete stability and quasi-optimality in L^p

Eliminates Gibbs phenomena in numerical solutions

Effective for 2-D advection problems

Abstract

We propose and analyse a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Lebesgue $L^p$-space, $1<p<\infty$. The greater generality of this weak setting is natural when dealing with rough data and highly irregular solutions, and when enhanced qualitative features of the approximations are needed. We first present a rigorous analysis of the well-posedness of the underlying continuous weak formulation, under natural assumptions on the advection-reaction coefficients. The main contribution is the study of several discrete subspace pairs guaranteeing the discrete stability of the method and quasi-optimality in $L^p$, and providing numerical illustrations of these findings, including the elimination of Gibbs phenomena, computation of optimal test spaces, and application to 2-D advection.