A Discontinuous Galerkin Reduced Basis Element Method For Elliptic Problems

We propose and analyse a new discontinuous reduced basis element method for the approximation of parametrized elliptic PDEs in partitioned domains. The method is built upon an offline stage (parameter independent) and an online (parameter dependent) one. In the offline stage we build a non-conforming (discontinuous) global reduced space as a direct sum of local basis functions generated independently on each subdomain. In the online stage, for any given value of the parameter, the approximate solution is obtained by ensuring the weak continuity of the fluxes and of the solution itself thanks to a discontinuous Galerkin approach. The new method extends and generalizes the methods introduced in [L. Iapichino, Ph.D. thesis, EPF Lausanne (2012); L. Iapichino, A. Quarteroni and G. Rozza, Comput. Methods Appl. Mech. Eng. 221-222 (2012) 63-82]. We prove its stability and convergence properties, as well as the spectral properties of the associated online algebraic system. We also propose a two-level preconditioner for the online problem which exploits the pre-existing decomposition of the domain and is based upon the introduction of a global coarse finite element space. Numerical tests are performed to verify our theoretical results.

Multi space reduced basis preconditioners for parametrized partial differential equations

Niccolò Dal Santo

The multiquery solution of parametric partial differential equations (PDEs), that is, PDEs depending on a vector of parameters, is computationally challenging and appears in several engineering contexts, such as PDE-constrained optimization, uncertainty quantification or sensitivity analysis. When using the finite element (FE) method as approximation technique, an algebraic system must be solved for each instance of the parameter, leading to a critical bottleneck when we are in a multiquery context, a problem which is even more emphasized when dealing with nonlinear or time dependent PDEs. Several techniques have been proposed to deal with sequences of linear systems, such as truncated Krylov subspace recycling methods, deflated restarting techniques and approximate inverse preconditioners; however, these techniques do not satisfactorily exploit the parameter dependence. More recently, the reduced basis (RB) method, together with other reduced order modeling (ROM) techniques, emerged as an efficient tool to tackle parametrized PDEs. In this thesis, we investigate a novel preconditioning strategy for parametrized systems which arise from the FE discretization of parametrized PDEs. Our preconditioner combines multiplicatively a RB coarse component, which is built upon the RB method, and a nonsingular fine grid preconditioner. The proposed technique hinges upon the construction of a new Multi Space Reduced Basis (MSRB) method, where a RB solver is built at each step of the chosen iterative method and trained to accurately solve the error equation. The resulting preconditioner directly exploits the parameter dependence, since it is tailored to the class of problems at hand, and significantly speeds up the solution of the parametrized linear system. We analyze the proposed preconditioner from a theoretical standpoint, providing assumptions which lead to its well-posedness and efficiency. We apply our strategy to a broad range of problems described by parametrized PDEs: (i) elliptic problems such as advection-diffusion-reaction equations, (ii) evolution problems such as time-dependent advection-diffusion-reaction equations or linear elastodynamics equations (iii) saddle-point problems such as Stokes equations, and, finally, (iv) Navier-Stokes equations. Even though the structure of the preconditioner is similar for all these classes of problems, its fine and coarse components must be accurately chosen in order to provide the best possible results. Several comparisons are made with respect to the current state-of-the-art preconditioning and ROM techniques. Finally, we employ the proposed technique to speed up the solution of problems in the field of cardiovascular modeling.

EPFL2018

A posteriori error estimation for the steady Navier-Stokes equations in random domains

Diane Sylvie Guignard, Fabio Nobile, Marco Picasso

We consider finite element error approximations of the steady incompressible Navier-Stokes equations defined on a randomly perturbed domain, the perturbation being small. Introducing a random mapping, these equations are transformed into PDEs on a fixed reference domain with random coefficients. Under suitable assumptions on the random mapping and the input data, in particular the so-called small data assumption, we prove the well-posedness of the problem. We assume then that the mapping depends affinely on L independent random variables and adopt a perturbation approach expanding the solution with respect to a small parameter ε that controls the amount of randomness in the problem. We perform an a posteriori error analysis for the first order approximation error, namely the error between the exact (random) solution and the finite element approximation of the first term in the expansion with respect to ε. Numerical results are given to illustrate the theoretical results and the effectiveness of the error estimators.

Elsevier2017

Multi space reduced basis preconditioners for parametrized partial differential equations

Niccolò Dal Santo

EPFL2018