A Petrov-Galerkin diffuse element method (PG DEM) and its comparison to EFG

The diffuse element method (DEM) pioneered by B. Nayroles, G. Touzot and P. Villon is a meshless method for solving partial differential equations. The original version does not pass the patch test. Here an improved version is proposed. This version is based on a Petrov-Galerkin formulation where test functions are required to meet different conditions than trial functions and is labeled here as PG DEM. It is shown that the original DEM derivative approximation is consistent but not integrable; therefore these functions are not suitable as test functions. A set of conditions on test and trial functions is applied. It is shown that the present Petrov-Galerkin (PG DEM) formulation satisfies the patch test and the numerical results show that it exhibits accuracy and rate of convergence superior to the original DEM, while the ''stiffness matrix'' will be unsymmetric even for self-adjoint problems.