Abstract
Abstract We propose and analyze 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 {L^{p}} -space, 1 < p < ∞ {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 {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.
Highlights
Residual minimization encapsulates the idea that an approximation to the solution u ∈ U of an operator equation Bu = f can be found by minimizing the norm of the residual f − Bwn amongst all wn in some finite-dimensional subspace Un ⊂ U
We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting
This powerful idea provides a stable and convergent discretization method under quite general assumptions, i.e., when B : U → V∗ is any linear continuous bijection from Banach space U onto the dual V∗ of a Banach space V, f ∈ V∗, and dist(u, Un) → 0 as n → ∞; see, e.g., Guermond [30, Section 2] for details. Note that this applies to well-posed weak formulations of linear partial differential equations (PDEs), in which case B is induced by the underlying bilinear form (i.e., ⟨Bw, v⟩ = b(w, v) for all w ∈ U and all v ∈ V)
Summary
The main contribution of this paper consists in the study of several elementary discrete subspace pairs (Un , Vm) for the DDMRes method for weak advection-reaction, including proofs of Fortin compatibility in the above-mentioned Banach-space setting. It thereby provides the first application and corresponding analysis of DDMRes in genuine (non-Hilbert) Banach spaces. This section contains several proofs of Fortin conditions, as well as some illustrative numerical examples pertaining to the Gibbs phenomena (Section 5.1), optimal test space basis (Section 5.2), and quasi-optimal convergence for 2-D advection (Section 5.3)
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