Discretisation of PDEs in Banach-space settings: Eliminating Gibbs phenomena and resolving non-Hilbert solutions
Is it possible to obtain near-best approximations to solutions of partial differential equations (PDEs) in a general Banach-space setting? Can this be done with guaranteed stability? I will address these questions by introducing the nonstandard, nonlinear Petrov-Galerkin (NPG) discretisation.
The NPG method is imperative for PDEs with rough data or nonsmooth solutions having discontinuities. Its theory generalises and extends Babuska’s theory for the classical Petrov-Galerkin method, as well as recent theories for residual-minimisation methods such as the discontinuous Petrov-Galerkin method (due to Demkowicz and Gopalakrishnan) and residual minimisation in Lp (due to Guermond). Crucial in the formulation of the NPG method is the nonlinear duality map, which is the natural extension of the Riesz map. I will show the stability of the NPG method and prove its quasi-optimality by extending a classical projection identity due to Kato.
To illustrate the significance of the new discretization framework, I will consider its application to the advection-reaction PDE and the Laplacian. Two of the main benefits of moving to Banach-space settings will be highlighted:
- The ability to eliminate the notorious Gibbs phenomena of numerical overshoots when the solution contains discontinuities.
- The ability to approximate on certain graded meshes, rough non-Hilbert solutions that can not be handled by the standard method in Hilbert spaces.
This is joint work with Ignacio Muga from Pontificia Universidad Catolica de Valparaiso.