Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids
In this paper, we consider anisotropic diffusion with decay, which takes the form alpha(x)c(x)-div[D(x)grad[c(x)]]=f(x) with decay coefficient alpha(x)>= 0, and diffusivity coefficient D(x) to be a second-order symmetric and positive-definite tensor. It is well known that this particular equation is a second-order elliptic equation, and satisfies a maximum principle under certain regularity assumptions. However, the finite element implementation of the classical Galerkin formulation for both anisotropic and isotropic diffusions with decay does not respect the maximum principle. Put differently, the classical Galerkin formulation violates the discrete maximum principle (DMP) for diffusion with decay even on structured computational meshes. We first show that the numerical accuracy of the classical Galerkin formulation deteriorates dramatically with an increase in alpha for isotropic media and violates the DMP. However, in the case of isotropic media, the extent of violation decreases with the mesh refinement. We then show that, in the case of anisotropic media, the classical Galerkin formulation for anisotropic diffusion with decay violates the DMP even at lower values of decay coefficient and does not vanish with mesh refinement. We then present a methodology for enforcing maximum principles under the classical Galerkin formulation for anisotropic diffusion with decay on general computational grids using optimization techniques. Representative numerical results (which take into account anisotropy and heterogeneity) are presented to illustrate the performance of the proposed formulation. Copyright (C) 2010 John Wiley & Sons, Ltd.
maximum principle general computational grids anisotropic diffusion with decay discrete maximum principle non-negative solutions convex quadratic programming