Minimal Residual Finite Element Methods in Banach Spaces

Abstract: Minimal-residual (MinRes) finite element methods have attracted significant attention in the recent numerical analysis literature, owing to their conceptual simplicity and striking stability properties. While these methods include classical least-squares and optimal Petrov-Galerkin methods, recent advances centre around the minimisation of residuals measured in a (discrete) dual norm, such as the discontinuous Petrov--Galerkin (DPG) methodology.

In this talk, I will first discuss how MinRes methods can be extended to Banach-space settings. This general setting allows for a direct discretization of PDEs in nonstandard non-Hilbert settings that are required when facing rough data and low-regular solutions. This development gives rise to a class of nonlinear Petrov--Galerkin methods, or, equivalently, abstract mixed methods with monotone nonlinearity. Discrete stability and quasi-optimal convergence follow under a Fortin condition. I will consider applications to PDEs (linear transport, advection-diffusion), as well as the regularization of rough linear functionals.

Secondly, I will show how the MinRes framework can be utilised for model reduction. In particular, I will present a machine-learning framework to train a provably stable parametric Petrov-Galerkin method on a fixed underlying mesh, whose aim is to ensure highly accurate quantities of interest regardless of the mesh size. Some recent numerics will illustrate these ideas.

computational engineering, finance, and sciencemathematical softwarenumerical analysiscomputational physics

Audience: researchers in the topic

Australian Seminar on Computational Mathematics

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