Mixed-hybrid finite elements and streamline computation for the potential flow problem.(English)Zbl 0767.76029
The authors discussed the application of a mixed-hybrid finite element method for solving equations of the form \(u=-A\nabla\varphi\), \(\nabla\cdot u=f\), where \(A\) is a symmetric and uniformly positive- definite second-rank tensor. The lowest-order mixed method is presented in detail. This equation is fundamental in the theory of heat conduction, electrostatics, and ground-water hydraulics.
The mixed finite element method results in a large system of linear equations. The choice of a numerical method to solve this system is restricted by the fact that its coefficient matrix is indefinite. This drawback can be circumvented by an implementation technique called hybridization, which leads to a sparse and symmetric positive-definite system of linear equations. This system can be solved efficiently by the preconditioned conjugate gradient method, where the preconditioning matrix is constructed by the incomplete Cholesky decomposition or the modified incomplete Cholesky decomposition. The applicability and advantages of the mixed finite element method and the efficient solution of the resulting system of linear equations are illustrated by several numerical experiments.
The benefits of the mixed method are apparent for problems with rough tensors of hydraulic conductivity and especially if the domain is subdivided into very flat subdomains. After an approximation, \(u\) has been computed by the mixed finite element method; streamlines and residence times can be determined efficiently and accurately using elementwise computations at the element level.
The mixed finite element method results in a large system of linear equations. The choice of a numerical method to solve this system is restricted by the fact that its coefficient matrix is indefinite. This drawback can be circumvented by an implementation technique called hybridization, which leads to a sparse and symmetric positive-definite system of linear equations. This system can be solved efficiently by the preconditioned conjugate gradient method, where the preconditioning matrix is constructed by the incomplete Cholesky decomposition or the modified incomplete Cholesky decomposition. The applicability and advantages of the mixed finite element method and the efficient solution of the resulting system of linear equations are illustrated by several numerical experiments.
The benefits of the mixed method are apparent for problems with rough tensors of hydraulic conductivity and especially if the domain is subdivided into very flat subdomains. After an approximation, \(u\) has been computed by the mixed finite element method; streamlines and residence times can be determined efficiently and accurately using elementwise computations at the element level.
Reviewer: C.T.Tsai
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 76S05 | Flows in porous media; filtration; seepage |
| 86A05 | Hydrology, hydrography, oceanography |
Keywords:
ground-water hydraulics;large system of linear equations;preconditioned conjugate gradient method;Cholesky decompositionCite
Full Text:DOI
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