A new mass lumping scheme for the mixed hybrid finite element method

Diffusion-type partial differential equation is a common mathematical model in physics. Solved by mixed finite elements, it leads to a system matrix which is not always an M-matrix. Therefore, the numerical solution may exhibit unphysical results due to oscillations. The criterion necessary to obtain an M-matrix is discussed in details for triangular, rectangular and tetrahedral elements. It is shown that the system matrix is never an M-matrix for rectangular elements and can be an M-matrix for trianaular an tetrahedral elements if criteria on the element's shape and on the time step length are fulfilled. A new mass lumping scheme is developed which leads to a less restrictive criterion: the discretization must be weakly acute (all angles less than pi/2) and there is no constraint on the time step length. The lumped formulation of mixed hybrid finite element can be applied not only to triangular meshes but also to more general shape elements in two and three dimensions. Numerical experiments show that, compared to the standard mixed hybrid formulation, the lumping scheme avoids (or strongly reduce) oscillations and does not create additional numerical errors. Copyright (c) 2006 John Wiley & Sons, Ltd.