A General Numerical Solution of the Two‐Dimensional Diffusion‐Convection Equation by the Finite Element Method

Abstract

The two-dimensional diffusion-convection equation, together with the appropriate auxiliary conditions, is used to describe approximately the motion of dissolved constituents in porous media flow, dispersion of pollutants in streams and estuaries, energy transfer in reservoirs, and other natural transport processes. The two-dimensional diffusion-convection equation, with an assumed set of auxiliary conditions, is converted to a variational principle for systems that do not involve mixed partials. The variational principle is in turn solved by the Ritz procedure by dividing the domain of interest into an arbitrary number of finite triangular elements. Within each element the unknown function states are represented by a first order space polynomial. The resulting system of first order linear equations is then solved by numerical differentiation using the Adams-Moulton multistep predictor-corrector method.

An Introduction to Meshfree Methods and Their Programming

“…This aspect of numerical treatment is addressed to certain degree of satisfaction in the finite difference method (FDM) by changing the interpolation scheme using so-called upwind grids (Courant et al, 1953;d Runchall and Wolfstein, 1969;Spalding 1972;Barrett, 1974;etc.). It has also been well studied by Guymon et al, (1970), Adey and Brebbia (1974), Zienkiewicz et al (1975), Christie et al (1976), Morton (1985), Donea et al, (1985), Hughes et al (1988), Onate (1998), and many others for the FEM. More detailed discussions on this issue can be found in the book by Zienkiewicz and Taylor (2000) and the references provided there.…”

Physically based modeling in catchment hydrology at 50: Survey and outlook

“…The connection between the governing equations for surface (river) flow, subsurface (groundwater) flow, and solute transport was made by Guymon [] and Guymon et al . [], who drew attention in particular to the common diffusion‐convection nature of the respective mass conservation equations for these domains. Early attempts at coupled hydrological models include those of Bresler [] and Bredehoeft and Pinder [] for flow and transport processes, Pinder and Sauer [] and Konikow and Bredehoeft [] for stream and aquifer dynamics, and Smith and Woolhiser [] for overland flow and soil infiltration.…”

Convection-Dominated Problems: Finite Element Approximations to the Convection-Diffusion-Reaction Equation

“…However, it was shown by Guymon et al [14] that it is a simple matter to derive a variational principle (or ensure self-adjointness, which is equivalent) if the operator is premultiplied by a suitable function p. Thus we write a weak form of Eq. However, it was shown by Guymon et al [14] that it is a simple matter to derive a variational principle (or ensure self-adjointness, which is equivalent) if the operator is premultiplied by a suitable function p. Thus we write a weak form of Eq.…”