Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids

arXiv:1003.5257v3 [cs.NA] 5 Jun 2010 ENFORCING THE NON-NEGATIVITY CONSTRAINT AND MAXIMUM PRINCIPLES FOR DIFFUSION WITH DECAY ON GENERAL COMPUTATIONAL GRIDS H. NAGARAJAN AND K. B. NAKSHATRALA Abstract. In this paper, we consider anisotropic diffusion with decay, which takes the form α(x)c(x) −div[D(x)grad[c(x)]] = f(x) with decay coefficient α(x) ≥0, and diffusivity coefficient D(x) to be a second-order symmetric and positive definite tensor. It is well-known that this partic- ular 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 for- mulation for both anisotropic and isotropic diffusion with decay does not respect the maximum principle. Put differently, the classical Galerkin formulation violates the discrete maximum princi- ple 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 α for isotropic media and violates the discrete maximum prin- ciple. 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 discrete maximum principle 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 numer- ical results (which take into account anisotropy and heterogeneity) are presented to illustrate the performance of the proposed formulation. 1. INTRODUCTION In this paper we consider heterogeneous anisotropic diffusion with decay, which takes the form: α(x)c(x)−div[D(x)grad[c(x)]] = f(x) with α(x) ≥0 and D(x) is a symmetric and positive definite second-order tensor. This equation is a linear second-order elliptic partial differential equation [21]. There are many important problems in Mathematical Physics which give rise to this equation [60]. Also, this equation arises in numerical methods and mathematical analysis of transient problems [35]. Some of these cases include: Date: November 17, 2018. Key words and phrases. maximum principle; discrete maximum principle; non-negative solutions; convex quadratic programming; anisotropic diffusion with decay; general computational grids. 1 (a) For certain gases, the diffusion process is accompanied by a decay of the molecules of the diffusing gas, and the decay is proportional to the concentration of the gas. Such a phenomenon can be modeled as a diffusion equation with decay. (b) For certain problems, the governing equation of diffusion in a moving domain can be trans- formed into a diffusion equation with decay. (c) Application of the method of horizontal lines to the transient diffusion equation (which is a linear parabolic partial differential equation) gives rise to a diffusion equation with decay. 1.1. Maximum principles and discrete maximum principles. From the theory of partial differential equations, it is well-known that the diffusion equation with decay satisfies a maximum principle under appropriate regularity assumptions. In some cases (but not always) the physically important condition that the concentration is non-negative is a direct consequence of a maximum principle. It is important to note that the classical maximum principle for diffusion with decay is different from the classical maximum principle for pure diffusion equation (see Theorem 2.1 and Remark 2.5 in this paper). It is imperative that predictive numerical simulations employ accurate and reliable discretization methods. The resulting discrete systems must inherit or mimic fundamental properties of contin- uous systems. The non-negative constraint and maximum principles are some of the essential properties of diffusion-type equations. However, it is well-known (and also discussed below) that many numerical formulations (including the popular ones) may not give non-negative solutions or satisfy maximum principles for these types of equations on general computational grids. Another point to note is that the satisfaction of maximum principles and the non-negative constraint by a numerical formulation will be altered by the presence of the decay term. (That is, the conditions under which a numerical formulation satisfies maximum principles and the non-negative constraint for pure diffusion can be different from those for diffusion with decay.) This leads us to discrete maximum principles. The discrete analogy of a maximum principle is commonly referred to as a discrete maximum principle (DMP). Some factors that affect discrete maximum principles are: numerical formulation, mesh size, element type, nature of the computational domain (e.g., presence/abse

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