Matrix-free subcell residual distribution for Bernstein finite elements: Low-order schemes and FCT

Abstract

In this work, we introduce a new residual distribution (RD) framework for the design of matrix-free bound-preserving finite element schemes. As a starting point, we consider continuous and discontinuous Galerkin discretizations of the linear advection equation. To construct the corresponding local extremum diminishing (LED) approximation, we perform mass lumping and redistribute the element residuals in a manner which guarantees the LED property. The hierarchical correction procedure for high-order Bernstein finite element discretizations involves localization to subcells and definition of bound-preserving weights for subcell contributions. Using strong stability preserving (SSP) Runge-Kutta methods for time integration, we prove the validity of discrete maximum principles under CFL-like time step restrictions. The low-order version of our method has roughly the same accuracy as the one derived from a piecewise (multi)-linear approximation on a submesh with the same nodal points. In high-order extensions, we currently use a flux-corrected transport (FCT) algorithm which can also be interpreted as a nonlinear RD scheme. The properties of the algebraically corrected Galerkin discretizations are illustrated by 1D, 2D, and 3D examples for Bernstein finite elements of different order. The results are as good as those obtained with the best matrix-based approaches. In our numerical studies for multidimensional problems, we use quadrilateral/hexahedral meshes but our methodology is readily applicable to unstructured/simplicial meshes as well.