The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach spaces
It provides a rigorous framework for approximating solutions in L^p spaces, which is beneficial for problems with rough data and irregular solutions, but the method is incremental as it extends existing minimal-residual techniques to a more general setting.
The paper proposes a minimal-residual method in discrete dual norms for solving advection-reaction equations in Banach spaces, achieving quasi-optimal approximations in L^p with elimination of Gibbs phenomena and stable discrete subspace pairs.
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.