On the finite volume reformulation of the mixed finite element method for elliptic and parabolic PDE on triangles

Abstract

A new resolution of parabolic and elliptic partial differential equations (PDEs) based on the mixed finite element approximation on triangles has been recently developed [24,25]. This new approach reduces the number of unknowns from fluxes or Lagrange multiplier defined on edges to a single unknown per element. In this paper, we analyze this transformation mathematically, and describe in details how to handle singular elements and singular edges. For these singular elements, the standard mixed method on triangles can always be made equivalent to a finite volume formulation, where the finite volumes are obtained by aggregation of finite elements across singular edges. The positive definiteness of the system matrix obtained with the new formulation is analyzed in details. A criterion is given concerning the property of this matrix which show that its conditioning is related to the shape of the triangle and the contrast in parameters from one element to the adjacent ones. Numerical experiments are performed for elliptic and parabolic PDEs. The comparisons between an iterative solver (PCG) and a direct solver (unifrontal/multifrontal) show that the direct solver is more efficient. Moreover, its performance is not correlated with the system matrix conditioning. It appears that the new formulation requires significantly less CPU time for elliptic PDEs and is competitive for parabolic PDEs. The new formulation remains also accurate enough even in nearly singular situations.