**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Publication# Interface control domain decomposition methods for heterogeneous problems

Abstract

This paper is concerned with the solution of heterogeneous problems by the interface control domain decomposition (ICDD) method, a strategy introduced for the solution of partial differential equations in computational domains partitioned into subdomains that overlap. After reformulating the original boundary value problem by introducing new additional control variables, the unknown traces of the solution at internal subdomain interfaces; the latter are determined by requiring that the (a priori) independent solutions in each subdomain undergo the minimization of a suitable cost functional.We provide an abstract formulation for coupled heterogeneous problems and a general theorem of well-posedness for the associated ICDD problem. Then, we illustrate and validate an efficient algorithm based on the solution of the Schur-complement system restricted solely to the interface control variables by considering two kinds of heterogeneous boundary value problems: the coupling between pure advection and advection-diffusion equations and the coupling between Stokes and Darcy equations. In the latter case, we also compare the ICDD method with a classical approach based on the Beavers-Joseph-Saffman conditions. Copyright (c) 2014 John Wiley & Sons, Ltd.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts

Loading

Related publications

Loading

Related publications (6)

Loading

Loading

Loading

Related concepts (15)

Problem solving

Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an a

Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.
The function is often thought of as

Darcy–Weisbach equation

In fluid dynamics, the Darcy–Weisbach equation is an empirical equation that relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid

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.

Simone Brugiapaglia, Fabio Nobile

We present a theoretical analysis of the CORSING (COmpRessed SolvING) method for the numerical approximation of partial differential equations based on compressed sensing. In particular, we show that the best s-term approximation of the weak solution of a PDE with respect to a system of N trial functions, can be recovered via a Petrov-Galerkin approach using m < N test functions. This recovery is guaranteed if the local a-coherence associated with the bilinear form and the selected trial and test bases fulfills suitable decay properties. The fundamental tool of this analysis is the restricted inf-sup property, i.e., a combination of the classical inf-sup condition and the well-known restricted isometry property of compressed sensing.

2018,

In this paper, we propose the mathematical and finite element analysis of a second-order partial differential equation endowed with a generalized Robin boundary condition which involves the Laplace--Beltrami operator by introducing a function space $H^1(\Omega; \Gamma)$ of $H^1(\Omega)$-functions with $H^1(\Gamma)$-traces, where $\Gamma \subseteq \partial \Omega$. Based on a variational method, we prove that the solution of the generalized Robin boundary value problem possesses a better regularity property on the boundary than in the case of the standard Robin problem. We numerically solve generalized Robin problems by means of the finite element method with the aim of validating the theoretical rates of convergence of the error in the norms associated to the space $H^1(\Omega; \Gamma)$.