6 research outputs found
The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces
© 2019 Walter de Gruyter GmbH, Berlin/Boston 2019. 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 Lp, 1 < p < ∞. 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
Eliminating Gibbs Phenomena: A Non-linear Petrov-Galerkin Method for the Convection-Diffusion-Reaction Equation
In this article we consider the numerical approximation of the
convection-diffusion-reaction equation. One of the main challenges of designing
a numerical method for this problem is that boundary layers occurring in the
convection-dominated case can lead to non-physical oscillations in the
numerical approximation, often referred to as Gibbs phenomena. The idea of this
article is to consider the approximation problem as a residual minimization in
dual norms in Lq-type Sobolev spaces, with 1 < q < . We then apply a
non-standard, non-linear PetrovGalerkin discretization, that is applicable to
reflexive Banach spaces such that the space itself and its dual are strictly
convex. Similar to discontinuous Petrov-Galerkin methods, this method is based
on minimizing the residual in a dual norm. Replacing the intractable dual norm
by a suitable discrete dual norm gives rise to a non-linear inexact mixed
method. This generalizes the Petrov-Galerkin framework developed in the context
of discontinuous Petrov-Galerkin methods to more general Banach spaces. For the
convection-diffusion-reaction equation, this yields a generalization of a
similar approach from the L2-setting to the Lq-setting. A key advantage of
considering a more general Banach space setting is that, in certain cases, the
oscillations in the numerical approximation vanish as q tends to 1, as we will
demonstrate using a few simple numerical examples
Gibbs Phenomena for -Best Approximation in Finite Element Spaces -- Some Examples
Recent developments in the context of minimum residual finite element methods
are paving the way for designing finite element methods in non-standard
function spaces. This, in particular, permits the selection of a solution space
in which the best approximation of the solution has desirable properties. One
of the biggest challenges in designing finite element methods are non-physical
oscillations near thin layers and jump discontinuities. In this article we
investigate Gibbs phenomena in the context of -best approximation of
discontinuities in finite element spaces with . Using carefully
selected examples, we show that on certain meshes the Gibbs phenomenon can be
eliminated in the limit as tends to . The aim here is to show the
potential of as a solution space in connection with suitably designed
meshes