Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains

TL;DR

This paper introduces advanced geometrically unfitted space-time finite element methods for PDEs on moving domains, achieving higher order accuracy in space and time through parametric mapping and robust stabilization techniques.

Contribution

It presents novel higher order unfitted space-time finite element methods with geometric accuracy and stabilization for PDEs on moving domains, validated through numerical experiments.

Findings

Achieved higher order accuracy in space and time.

Demonstrated robustness with ghost penalty stabilization.

Validated methods across different dimensions and polynomial degrees.

Abstract

In this paper, we propose new geometrically unfitted space-time Finite Element methods for partial differential equations posed on moving domains of higher order accuracy in space and time. As a model problem, the convection-diffusion problem on a moving domain is studied. For geometrically higher order accuracy, we apply a parametric mapping on a background space-time tensor-product mesh. Concerning discretisation in time, we consider discontinuous Galerkin, as well as related continuous (Petrov-)Galerkin and Galerkin collocation methods. For stabilisation with respect to bad cut configurations and as an extension mechanism that is required for the latter two schemes, a ghost penalty stabilisation is employed. The article puts an emphasis on the techniques that allow to achieve a robust but higher order geometry handling for smooth domains. We investigate the computational properties of the respective methods in a series of numerical experiments. These include studies in different dimensions for different polynomial degrees in space and time, validating the higher order accuracy in both variables.