Roman Stuhlmacher: Smoothness Increasing Accuracy Conserving filtering on nonuniform meshes

Abstract:

Smoothness Increasing Accuracy Conserving (SIAC) filtering is a versatile post-processing technique designed to enhance the accuracy and smoothness of numerical solutions to differential equations. By convolving the solution of a finite element scheme with a carefully constructed kernel, SIAC filtering achieves both higher accuracy and improved smoothness in the post-processed solution. Initially developed in the context of Lagrange finite elements for elliptic equations, the SIAC framework has been extended to parabolic and hyperbolic problems and other discretization methods, most notably Discontinuous Galerkin (DG) methods. However, its reliance on uniform meshes remains a key limitation, restricting its use in more complex applications. My current research seeks to overcome this limitation by developing a robust theoretical framework for SIAC filtering on nonuniform meshes. In this presentation, I will provide an introduction to SIAC filtering, explore the challenges associated with nonuniform meshes, and discuss approaches to address these issues.