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Texas A&M University
Mathematics

Numerical Analysis Seminar

Date: November 9, 2022

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Vladimir Yushutin, Clemson University

  

Title: T-Rex FEM: an abstract analysis framework for unfitted methods

Abstract: Unfitted, non-conforming finite element methods have the following in common: there is a drastic difference between the space of solutions and the finite element space. This difference manifests on the discrete level where one needs to employ a discrete stabilization form to guarantee the well-posedness of linear problems. Convergence analysis for such methods often follows the second Strang lemma, conditions of which may be hard to verify in some situations.

Instead, we study the strong convergence of unfitted continuous-in-time approximations via compactness. With this goal in mind, we develop an analysis framework, called T-Rex FEM, that involves notions of abstract TRace and EXtension operators. We build this analysis framework sequentially starting from abstract linear elliptic, parabolic, saddle problems and applying it to Navier--Stokes and Allen--Cahn equations.

The key ingredient is a problem-dependent modification of the abstract discrete stabilization form that makes the scheme amenable to a proof by compactness. We test numerically the modified scheme suggested by the T-Rex FEM when it is applied to the surface heat equation being solved by the Trace FEM - an unfitted method for surface PDEs which uses a bulk mesh surrounding the surface. In addition to the advantage of T-Rex FEM from the analysis standpoint, the new scheme restores the conditioning of linear problems, known in the fitted case for the heat equation, despite the presence of the stabilization form.