Domain decomposition preconditioners

The purpose of this thesis is to define efficient parallel preconditioners based on the domain decomposition paradigm and to apply them to the solution of the steady compressible Euler equations. In the first part we propose and analyse various domain decomposition preconditioners of both overlapping (Schwarz) and non-overlapping (Schur complement-based) type. For the former, we deal with two-level methods, with an algebraic formulation of the coarse space. This approach enjoys several interesting properties not always shared by more standard twolevel methods. For the latter, we introduce a class of preconditioners based on a peculiar decomposition of the computational domain. The domain is decomposed in such a way that one subdomain is connected to all the others, which are in fact disconnected components. A class of approximate Schur complement preconditioners is also considered. Theoretical and numerical results are given for a model problem. In the second part we consider the application of the previous domain decomposition preconditioners to the compressible Euler equations. The discretisation procedure, based on multidimensional upwind residual distribution schemes, is outlined. We introduce a framework that combines non-linear system solvers, Krylov accelerators, domain decomposition preconditioners, as well as mesh adaptivity procedures. Several numerical tests of aeronautical interest are carried out in order to assess both the discretisation schemes and the mesh adaptivity procedures. In the third part we consider the parallel aspects inherent in the numerical solution of the compressible Euler equations on parallel computers with distributed memory. All the main kernels of the solution algorithm are analysed. Many numerical tests are carried out, with the aim of investigating the performance of the domain decomposition preconditioners proposed in the first part of the thesis, in the applications addressed in the second part.

Object-oriented finite element programming

New technologies in computer science applied to numerical computations open the door to alternative approaches to mechanical problems using the finite element method. In classical approaches, theoretical developments often become cumbersome and the computer model which follows shows resemblance with the initial problem statement. The first step in the development consists usually in the analysis of the physics of the problem to simulate. The problem is generally described by a set of equations including partial differential equations. This first model is then replaced by successive equivalent or approximated models. The final result consists in a mathematical description of elemental matrices and algorithms describing the matrix form of the problem. The traditional approach consists then in constructing a computer model, generally complex and often quite different from the original mathematical description, thus making further corrections difficult. Therefore, the crucial problem of both the software architecture and the choice of the appropriate programming language is raised. Partially breaking with this approach, we propose a new approach to develop and program finite element formulations. The approach is based on a hybrid symbolic/numerical approach on the one hand, and on a high level software tool, object-oriented programming (supported here by the languages Smalltalk and C++) on the other hand. The aim of this work is to develop an appropriate environment for the algebraic manipulations needed for a finite element formulation applied to an initial boundary value problem, and also to perform efficient numerical computations. The new environment should make it possible to manage al1 the concepts necessary to solve a physical problem: manipulation of partial differential equations, variational formulations, integration by parts, weak forms, finite element approximations… The concepts manipulated therefore remain closely related to the original mathematical framework. The result of these symbolic manipulations is a set of elemental data (mass matrix, stiffness matrix, tangent stiffness matrix,…) to be introduced in a classical numerical code. The object-oriented paradigm is essential to the success of the implementation. In the context of the finite element codes, the object-oriented approach has already proved its capacity to represent and handle complex structures and phenomena. This is confirmed here with the symbolic environment for derivation of finite element formulations in which objects such as expression, integral and variational formulation appear. The link between both the numerical world and the symbolic world is based on an object-oriented concept for automatic programmation of matrix forms derived from the finite element method. As a result, a global environment in which the numerical is capable of evolving, using a language close to the natural mathematical one, is achieved. The potential of the approach is further demonstrated, on the one hand, by the wide range of problems solved in linear mechanics (electrodynamics in 1 and 2D, heat diffusion,…) as well as in nonlinear mechanics (advection dominated 1D problem, Navier Stokes problem), and, on the other hand by the diversity of the formulations manipulated (Galerkin formulations, space-time Galerkin formulations continuous in space and discontinuous in time, generalized Galerkin least-squares formulations).

EPFL1997

Adaptive WENO methods based on radial basis functions reconstruction

Caterina Bigoni, Jan Sickmann Hesthaven

We explore the use of radial basis functions (RBF) in the weighted essentially non-oscillatory (WENO) reconstruction process used to solve hyperbolic conservation laws, resulting in a numerical method of arbitrarily high order to solve problems with discontinuous solutions. Thanks to the mesh-less property of the RBFs, the method is suitable for non uniform grids and mesh adaptation. We focus on multiquadric radial basis functions and propose a simple strategy to choose the inherent shape parameter to control the balance between theoretical achievable accuracy and the numerical stability. We also develop an original smoothness indicator independent of the chosen RBF for the WENO reconstruction step. Moreover, we introduce type I and type II RBF-WENO methods by computing specific linear weights. The RBF-WENO method is used to solve linear and nonlinear problems for both scalar conservation laws and systems of conservation laws, including Burgers equation, the Buckley-Leverett equation, and the Euler equations. Numerical results confirm the performance of the proposed method. We finally consider an effective conservative adaptive algorithm that captures moving shocks and rapidly varying solutions well. Numerical results on moving grids are presented for both Burgers equation and the more complex Euler equations.

Springer Verlag2017

Substructured Two-grid and Multi-grid Domain Decomposition Methods

Tommaso Vanzan

Two-level domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume. In this paper, we use the excellent smoothing properties of domain decomposition methods to define a new class of substructured two-level methods, for which both domain decomposition smoothers and coarse correction steps are defined on the interfaces. This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, our approach does not require the explicit construction of coarse spaces, and it permits a multilevel extension, which is desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

2021

Object-oriented finite element programming

EPFL1997