arXiv subjects

Frederik Schnack

Publications and source records attributed to Frederik Schnack.

3 recordsLinked to original sources

arXiv2025

Broken-FEEC on multipatch domains with local refinements

This article introduces a novel approach for broken-FEEC (Finite Element Exterior Calculus), extending its application to locally refined spline spaces with non-matching interfaces. Traditional broken-FEEC allows for discontinuous discretizations at patch interfaces, preserving the de Rham structure and offering computational benefits. However, local refinements often lead to numerical artifacts. Our solution involves developing moment-preserving discrete conforming projection operators. These operators are explicit, localized, and metric-independent, ensuring $H^1$ and $H(\text{curl})$ continuity across non-matching interfaces while preserving high-order polynomial moments. This results in broken-FEEC de Rham sequences with accurate strong and weak derivatives, leading to energy-preserving Maxwell solvers that are explicit and virtually free of spurious modes. Numerical simulations confirm the efficacy of our method in eliminating spurious waves.

math.NA↗

arXiv2024

Splitting scheme for gyro-kinetic equations with Semi-Lagrangian and Arakawa substeps

The gyro-kinetic model is an approximation of the Vlasov-Maxwell system in a strongly magnetized magnetic field. We propose a new algorithm for solving it combining the Semi-Lagrangian (SL) method and the Arakawa (AKW) scheme with a time-integrator. Both methods are successfully used in practice for different kinds of applications, in our case, we combine them by first decomposing the problem into a fast (parallel) and a slow (perpendicular) dynamical system. The SL approach and the AKW scheme will be used to solve respectively the fast and the slow subsystems. Compared to the scheme in [1], where the entire model is solved using only the SL method, our goal is to replace the method used in the slow subsystem by the AKW scheme, in order to improve the conservation of the physical constants.

math.NA↗

arXiv2023

Bounded commuting projections for multipatch spaces with non-matching interfaces

We present commuting projection operators on de Rham sequences of two-dimensional multipatch spaces with local tensor-product parametrization and non-matching interfaces. Our construction yields projection operators which are local and stable in any $L^p$ norm with $p \in [1,\infty]$: it applies to shape-regular spline patches with different mappings and local refinements, under the assumption that neighboring patches have nested resolutions and that interior vertices are shared by exactly four patches. It also applies to de Rham sequences with homogeneous boundary conditions. Following a broken-FEEC approach, we first consider tensor-product commuting projections on the single-patch de Rham sequences, and modify the resulting patch-wise operators so as to enforce their conformity and commutation with the global derivatives, while preserving their projection and stability properties with constants independent of both the diameter and inner resolution of the patches.

math.NA↗