Sarah Roggendorf

University of Cambridge, UK

We consider the numerical approximation of partial differential equations whose solutions may contain sharp features, such as interior and boundary layers. One of the main challenges of designing a numerical method for these types of problems is that these sharp features can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena.

The idea we are pursuing is to consider the approximation problem as a residual minimization in dual norms in L^q-type Sobolev spaces, with 1<q<2. We then apply a non-standard, non-linear Petrov-Galerkin discretization, proposed in [1] that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous Petrov-Galerkin methods [2], this method is based on employing optimal test functions. Replacing the intractable optimal test space by a suitable computable approximation gives rise to a non-linear inexact mixed method for which optimal a priori estimates hold. This generalizes the Petrov-Galerkin framework developed in the context of discontinuous Petrov-Galerkin methods to more general Banach spaces. For the convection-diffusion equation, this yields a generalization of the approach described in [3] from the L^2-setting to the L^q-setting.

The choice of the discrete spaces relies on results regarding the L^q-best approximation of discontinuities in finite element spaces with 1 ≤ q < ∞. We show that on certain meshes the Gibbs phenomenon in the L^q-best approximation can be eliminated in the limit as q tends to 1. Furthermore, we demonstrate that these results can be used to design discrete spaces for our non-linear Petrov-Galerkin scheme in order to eliminate non-physical oscillations in the approximation of boundary- and interior layers in the solution of convection-diffusion equations.