A multi-dimensional monotonic finite element model for solving the convection-diffusion-reaction equation. (English) Zbl 1014.76040
Summary: We develop a finite element model for solving convection-diffusion-reaction equation in two dimensions with the aim to enhance the scheme stability without compromising consistency. Reducing errors of false diffusion type is achieved by adding an artificial term to get rid of three leading mixed derivative terms in Petrov-Galerkin formulation. The finite element model of Petrov-Galerkin type, while maintaining convective stability, is modified to suppress oscillations about the sharp layer by employing the \(M\)-matrix theory. To validate this monotonic model, we consider test problems which are amenable to analytic solutions. Good agreement is obtained in both one- and two-dimensional problems, thus validating the method. Other problems suitable for benchmarking the proposed model are also investigated.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76R99 | Diffusion and convection |
Keywords:
convection-diffusion-reaction equation; scheme stability; Petrov-Galerkin formulation; \(M\)-matrix theory; monotonic model; artificial term; finite element model; convective stabilityReferences:
| [1] | Ilinca, AIAA Journal 36 pp 44– (1988) |
| [2] | Numerical Simulation of Non-Newtonian Flow. Elsevier: New York, 1984. · Zbl 0583.76002 |
| [3] | Leboucher, Journal of Computational Physics 150 pp 181– (1999) · Zbl 0935.76050 |
| [4] | Harari, Computer Methods in Applied Mechanics and Engineering 87 pp 59– (1991) |
| [5] | Ataie-Ashtini, Journal of Contaminant Hydrology 23 pp 149– (1996) |
| [6] | Hossain, Applied Mathematics and Computation 102 pp 101– (1999) |
| [7] | Hossain, Applied Mathematics and Computation 100 pp 249– (1999) |
| [8] | Harari, Computer Methods in Applied Mechanics and Engineering 155 pp 165– (1994) |
| [9] | Idelsohn, Computer Methods in Applied Mechanics and Engineering 136 pp 27– (1996) |
| [10] | Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Prentice-Hall: New Jersey, 1998; 454-456. |
| [11] | Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press: Dubin, 1980. · Zbl 0459.65058 |
| [12] | Tezdugar, Computer Methods in Applied Mechanics and Engineering 59 pp 307– (1986) |
| [13] | A chock-capturing anisotropic diffusion for the finite element solution of the diffusion-convection-reaction equation. Finite Element in Fluids, New trends and Applications Part 1, (ed.), Pineridge: Swansea, 1993; 67-75. |
| [14] | Codina, Computer Methods in Applied Mechanics and Engineering 156 pp 185– (1998) |
| [15] | Warming, Journal of Computational Physics 14 pp 159– (1974) |
| [16] | A multi-dimensional upwind scheme with no crosswind diffusion. In Finite Element Methods for Convection Dominated Flows, (ed.), AMD 34: ASME: New York, 1979; 19-35. |
| [17] | Numerical solution of partial differential equations. In Applied Mathematical Science. (eds). Springer: Berlin, 1981; 32. |
| [18] | Linß, Computer Methods in Applied Mechanics and Engineering 190 pp 3527– (2001) |
| [19] | In Finite Element for Convection Dominated Flows, (ed.), in AMD 34. ASME: New York, 1979; 91-104. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
