Double-grid quadrature with interpolation-projection (DoGIP) as a novel discretisation approach: An application to FEM on simplexes

TL;DR

This paper introduces DoGIP, a novel matrix-free discretisation method for FEM on simplexes that reduces memory usage at the cost of increased computation, demonstrated through heat transfer and elliptic problem examples.

Contribution

It generalizes the DoGIP approach from Fourier--Galerkin to FEM on simplexes, enabling efficient matrix-free discretisation with potential for further research.

Findings

Reduces memory demands for higher-order basis functions

Applicable to weighted projection and elliptic problems

Open questions for further investigation

Abstract

This paper is focused on the double-grid integration with interpolation-projection (DoGIP), which is a novel matrix-free discretisation method of variational formulations introduced for Fourier--Galerkin approximation. Here, it is described as a more general approach with an application to the finite element method (FEM) on simplexes. The approach is based on treating the trial and a test function in variational formulation together, which leads to the decomposition of a linear system into interpolation and (block) diagonal matrices. It usually leads to reduced memory demands, especially for higher-order basis functions, but with higher computational requirements. The numerical examples are studied here for two variational formulations: weighted projection and scalar elliptic problem modelling, e.g. diffusion or stationary heat transfer. This paper also opens a room for further investigation, which is discussed in the conclusion.