We discuss the general energetic variational approaches for hydrodynamic systems of complex fluids. In these energetic variational approaches, the least action principle (LAP) with action functional gives the Hamiltonian parts (conservative force) of the hydrodynamic systems, and the maximum/minimum dissipation principle (MDP), i.e., Onsager's principle, gives the dissipative parts (dissipative force) of the systems. When we combine the two systems derived from the two different principles, we obtain a whole coupled nonlinear system of equations satisfying the dissipative energy laws. We will discuss the important roles of MDP in designing numerical method for computations of hydrodynamic system in complex fluids. We will reformulate the dissipation in energy equation in terms of a rate in time by using an appropriate evolution equations, then the MDP is employed in the reformulated dissipation to obtain the dissipative force for the hydrodynamic system. The systems are consistent with the Hamiltonian parts which are derived from LAP. This procedure allows the usage of lower order element (a continuous C 0 finite element) in numerical method to solve the system rather than high order elements, and at the same time preserves the dissipative energy law. We also verify this method through some numerical experiments in simulating the free interface motion in the mixture of two different fluids.
In this paper, we present an analysis of a multigrid method for nonsymmetric and/or indefinite elliptic problems.In this multigrid method various types of smoothers may be used. One type of smoother which we consider is defined in terms of an associated symmetric problem and includes point and line, Jacobi and Gauss- Seidel iterations.We also study smoothers based entirely on the original operator.One is based on the normal form, that is, the product of the operator and its transpose, Other smoothers studied include point and line, Jacobi and Gauss-Seidel.We show that the uniform estimates of (ref. 6) for symmetric positive definite problems carry over to these algorithms. More precisely, the multigrid iteration for the nonsymmetric and/or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine (but not depending on the number of multigrid levels).
Abstract. The finite difference multigrid solution of an optimal control problem associated with an elliptic equation is considered. Stability of the finite difference optimality system and optimalorder error estimates in the discrete L 2 norm and in the discrete H 1 norm under minimum smoothness requirements on the exact solution are proved. Sharp convergence factor estimates of the two grid method for the optimality system are obtained by means of local Fourier analysis. A multigrid convergence theory is provided which guarantees convergence of the multigrid process towards weak solutions of the optimality system. Key words. optimal control problem, Poisson equation, finite differences, accuracy estimate, convergence theory, multigrid method AMS subject classifications. 49K20, 65N06, 65N12, 65N55PII. S03630129013934321. Introduction. Optimal control problems involving partial differential equations [17,18] are nowadays receiving much attention because of their importance in the industrial design process. Especially, the need for accurate and efficient solution methods for these problems has become an important issue.We consider a finite difference framework and multigrid methods for the case of distributed optimal control of an elliptic problem and provide for this case optimal estimates for the accuracy of the solution and for the convergence factor of the multigrid process. The present work is characterized by the fact that we extend known analytic tools for scalar elliptic problems to the case of a (nonsymmetric) system of elliptic partial differential equations, called an optimality system.In our finite difference analysis, based on results stated in [14,20], we prove stability of the finite difference optimality system and prove optimal-order error estimates in the discrete L 2 norm and in the discrete H 1 norm under minimum smoothness requirements on the analytic solution.It is known that multigrid methods [5,13,21] solve elliptic problems with optimal computational order, i.e., the number of computer operations required scales linearly with respect to the number of unknowns. This fact has been demonstrated in the case of multigrid applied to a singular optimal control problem associated with a nonlinear elliptic equation [2]. In particular, results in [2] show that the convergence properties of the multigrid method do not deteriorate as the weight of the cost of the control tends to zero, demonstrating the robustness of this method.We prove convergence of the multigrid method applied to the optimality system within two analytic frameworks which have complementary features. We use two grid
We consider a covolume or finite volume method for a system of first-order PDEs resulting from the mixed formulation of the variable coefficient-matrix Poisson equation with the Neumann boundary condition. The system may represent either the Darcy law and the mass conservation law in anisotropic porous media flow, or Fourier law and energy conservation. The velocity and pressure are approximated by the lowest order Raviart-Thomas space on triangles. We prove its first-order optimal rate of convergence for the approximate velocities in the L 2 -and H(div; Ω)-norms as well as for the approximate pressures in the L 2 -norm. Numerical experiments are included.
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