A Conservative Flux Optimization Finite Element Method for Convection-diffusion Equations

Abstract

This article presents a high order conservative flux optimization (CFO) finite element method for the elliptic diffusion equations. The numerical scheme is based on the classical Galerkin finite element method enhanced by a flux approximation on the boundary of a prescribed set of arbitrary control volumes (either the finite element partition itself or dual voronoi mesh, etc). The numerical approximations can be characterized as the solution of a constrained-minimization problem with constraints given by the flux conservation equations on each control volume. The discrete linear system is a typical saddle-point problem, but with less number of degrees of freedom than the standard mixed finite element method, particularly for elements of high order. Moreover, the numerical solution of the proposed scheme is of super-closeness with the finite element solution. Error estimates of optimal order are established for the numerical flux as well as the primary variable approximations. We present several numerical studies in order to verify convergence of the CFO schemes. A simplified two-phase flow in highly heterogeneous porous media model problem will also be presented. The numerical results show obvious advantages of applying high order CFO schemes.

Superconvergence of the gradient approximation for weak Galerkin finite element methods on nonuniform rectangular partitions

“…Here the value of the coefficient α x i , α y i , α i are specified in Table 7.22. This test problem has been considered in [16]. The numerical results are shown in Table 7.23.…”

“…The exact solution for this test problem is given by u = 1−2y 2 +4xy +6x+2y for x < 0.5 and u = −2y 2 + 1.6xy − 0.6x + 3.2y + 4.3 for x ≥ 0.5. This test problem has been considered in [16]. The numerical results are illustrated in Table 7.24.…”

Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem

“…For a related method using finite element spaces with C 1 -regularity see [22] and for methods designed for well-posed, but indefinite problems, we refer to [9] and for second order elliptic problems on non-divergence form see [38] and [39]. Recently approaches similar to those discussed in this work were proposed for the approximation of well-posed convection-diffusion problems [27] or porous media flows [32].…”

Primal Dual Mixed Finite Element Methods for Indefinite Advection-Diffusion Equations

“…Similar ideas have recently been exploited successfully in the context of weak Galerkin methods for elliptic problems on non-divergence form [40], Fokker-Planck equations [39], and the ill-posed elliptic Cauchy problem in [41]. In [35] a method was introduced which is reminiscent of the lowest order version of the method we propose herein. The case of high Peclet number was, however, not considered in [35], so our analysis is likely to be useful for the understanding of the method in [35] in this regime.…”

“…In [35] a method was introduced which is reminiscent of the lowest order version of the method we propose herein. The case of high Peclet number was, however, not considered in [35], so our analysis is likely to be useful for the understanding of the method in [35] in this regime.…”