The mixed-cell-complex partition-of-unity method

We present a Galerkin method for solving partial differential equations which is a blend of ideas from particle-based methods on the one side and traditional finite element methods on the other side. The method is here named mixed-cell-complex partition of unity method (MCCPUM). It can be arbitrarily considered as being based on a set of scattered particles in the domain and on its boundary, or on a Delaunay cell decomposition of the domain. In contrast to the element-free Galerkin method and other meshless techniques, the partition of unity is not constructed on circular or rectangular supports, but rather on overlapping polyhedral supports generated from a Voronoi/Delaunay decomposition of space: the mixed-cell-complex. This approach inherits most of the advantages of truly meshless schemes, while it greatly facilitates the numerical integration of the weak forms required in Galerkin approximations. The discretization is exclusively governed by the selection of nodes and the approximation orders associated to the nodes locally. Here Legendre polynomials of arbitrary orders are used. The mixed-cell-complex and the corresponding Galerkin discretization are explained, numerical examples for the Poisson problem in two dimensions are presented, and the efficiency of the method is discussed.