A Domain Decomposition Method Based on Weighted Interior Penalties for Advection-Diffusion-Reaction Problems

We propose a domain decomposition method for advection-diffusion- reaction equations based on Nitsche's transmission conditions. The advection-dominated case is stabilized using a continuous interior penalty approach based on the jumps in the gradient over element boundaries. We prove the convergence of the finite element solutions of the discrete problem to the exact solution and propose a parallelizable iterative method. The convergence of the resulting domain decomposition method is proved, and this result holds true uniformly with respect to the diffusion parameter. The numerical scheme that we propose here can thus be applied straightforwardly to diffusion-dominated, advection-dominated, and hyperbolic problems. Some numerical examples are presented in different flow regimes showing the influence of the stabilization parameter on the performance of the iterative method, and we compare our method with some other domain decomposition techniques for advection-diffusion equations.

Finite element approximation of multi-scale elliptic problems using patches of elements

Jacques Rappaz

In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679-684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presented.In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679-684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presented.

Springer-Verlag2005

Stabilized reduced basis method for parametrized advection-diffusion PDEs

Paolo Pacciarini

The aim of this master thesis is to study and develop a stabilized reduced basis method suitable for the approximation of the solution of parametrized advection-diffusion PDEs with high Péclet number, that is, roughly, the ratio between the advection coefficient and the diffusion one. Advection-diffusion equations are very important in many engineering applications, because they are used to model, for example, heat transfer phenomena or the diffusion of pollutants in the atmosphere. In such applications, we often need very fast evaluations of the approximated solution, depending on some input parameters. This happens, for example, in the case of real-time simulation. Moreover, we need rapid evaluations also if we have to perform repeated approximation of the solution, for different input parameters. An important case of this many-query situation is represented by some optimization problems, in which the objective function to optimize depends on the parameters through the solution of a PDE. The reduced basis (RB) method meets our need for rapidity and it is also able to guarantee the reliability of the solution, thanks to sharp a posteriori error bounds. We can find in literature many works about the application of the RB method to advection-diffusion problems but they mainly deal with equations in which the Péclet number is low. The need for stabilization arise from the fact that the finite element (FE) approximated solution - that the RB method aims to recover - shows strong instability problems that have to be fixed. In this work we want to go further in the study of the stabilization of the RB method for advection dominated problem in both steady and unsteady case. As regards the steady case, we first compare two possible stabilization strategies, by testing them on some test problems, in order to design an efficient stabilized reduced basis method. We will then test this method using the piecewise quadratic FE approximation as reference solution, instead of the usual piecewise linear one. We extend the method designed for the steady case to the time dependent case and we will carry out some numerical tests.

2012

MATHICSE Technical Report : A numerical investigation of multi space reduced basis preconditioners for parametrized elliptic advection-diffusion

Niccolò Dal Santo, Simone Deparis, Andrea Manzoni

We analyze the numerical performance of a preconditioning technique recently proposed in [4] for the ecient solution of parametrized linear systems arising from the finite element (FE) discretization of parameter-dependent elliptic partial differential equations (PDEs). In order to exploit the parametric dependence of the PDE, the proposed preconditioner takes advantage of the reduced basis (RB) method within the preconditioned iterative solver employed to solve the linear system, and combines a RB solver, playing the role of coarse component, with a traditional fine grid (such as Additive Schwarz or block Jacobi) preconditioner. A sequence of RB spaces is required to handle the approximation of the error-residual equation at each step of the iterative method at hand, whence the name of Multi Space Reduced Basis (MSRB) method. In this paper, a numerical investigation of the proposed technique is carried on in the case of a Richardson iterative method, and then extended to the flexible GMRES method, in order to solve parameterized advection-diffusion problems. Particular attention is payed to the impact of anisotropic diffusion coeffcients and (possibly dominant) transport terms on the proposed preconditioner, by carrying out detailed comparisons with the current state of the art algebraic multigrid preconditioners.

MATHICSE2017

Stabilized reduced basis method for parametrized advection-diffusion PDEs

Paolo Pacciarini

2012