  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.
