Space Decompositions and Solvers for Discontinuous Galerkin Methods

We present a brief overview of the different domain and space decomposition techniques that enter in developing and analyzing solvers for discontinuous Galerkin methods. Emphasis is given to the novel and distinct features that arise when considering DG discretizations over conforming methods. Connections and differences with the conforming approaches are emphasized.Comment: 2 pages 2 figures no table

An agglomeration-based massively parallel non-overlapping additive Schwarz preconditioner for high-order discontinuous Galerkin methods on polytopic grids

In this article we design and analyze a class of two-level non-overlapping additive Schwarz preconditioners for the solution of the linear system of equations stemming from discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polytopic meshes. The preconditioner is based on a coarse space and a non-overlapping partition of the computational domain where local solvers are applied in parallel. In particular, the coarse space can potentially be chosen to be non-embedded with respect to the finer space; indeed it can be obtained from the fine grid by employing agglomeration and edge coarsening techniques. We investigate the dependence of the condition number of the preconditioned system with respect to the diffusion coefficient and the discretization parameters, i.e., the mesh size and the polynomial degree of the fine and coarse spaces. Numerical examples are presented which confirm the theoretical bounds

Robust and parallel scalable iterative solutions for large-scale finite cell analyses

The finite cell method is a highly flexible discretization technique for numerical analysis on domains with complex geometries. By using a non-boundary conforming computational domain that can be easily meshed, automatized computations on a wide range of geometrical models can be performed. Application of the finite cell method, and other immersed methods, to large real-life and industrial problems is often limited due to the conditioning problems associated with these methods. These conditioning problems have caused researchers to resort to direct solution methods, which signifi- cantly limit the maximum size of solvable systems. Iterative solvers are better suited for large-scale computations than their direct counterparts due to their lower memory requirements and suitability for parallel computing. These benefits can, however, only be exploited when systems are properly conditioned. In this contribution we present an Additive-Schwarz type preconditioner that enables efficient and parallel scalable iterative solutions of large-scale multi-level hp-refined finite cell analyses.Comment: 32 pages, 17 figure

Solution of the 2D Navier-Stokes equations by a new FE fractional step method.

In this work, a mathematical and numerical approach for the solution of the 2D Navier-Stokes equations for incompressible fluid flow problems is investigated. A new flux conservative technique for the solution of the elliptic part of the equations is formulated. In the new model, the non linear convective terms of the momentum equations are approximated by means of characteristics and the spatial approximations, of equal order, are obtained by polynomials of degree two. The advancing in time is afforded by a fractional step method combined with a suitable stabilization technique so that the Inf-Sup condition is respected. In order to keep down the computational cost, the algebraic systems are solved by an iterative solver (Bi-CGSTAB) preconditioned by means of Schwarz additive scalable preconditioners. The properties of the new method are verified carrying out several numerical tests. At first, some elliptic, parabolic and convective-diffusive problems are solved and discussed, then the results of some time dependent and stationary 2D Navier-Stokes problems (in particular the well known benchmark problem of the natural convection in a square cavity) are discussed and compared to those found in the literature. Another, potentially very important application of the numerical tools developed, regards the solution of 1D Shallow-Water equations. In fact the use of the fractional steps scheme for advancing in time and the finite elements (of different polynomial degrees) for the spatial approximation, makes the above mentioned approach computationally profitable and convenient for real applications. The efficiency and accuracy of the numerical model have been checked by solving a theoretical test. Finally, a brief description of the software suitably developed and used in the tests conclude the thesis

Solution of the 2D Navier-Stokes equations by a new FE fractional step method.

In this work, a mathematical and numerical approach for the solution of the 2D Navier-Stokes equations for incompressible fluid flow problems is investigated. A new flux conservative technique for the solution of the elliptic part of the equations is formulated. In the new model, the non linear convective terms of the momentum equations are approximated by means of characteristics and the spatial approximations, of equal order, are obtained by polynomials of degree two. The advancing in time is afforded by a fractional step method combined with a suitable stabilization technique so that the Inf-Sup condition is respected. In order to keep down the computational cost, the algebraic systems are solved by an iterative solver (Bi-CGSTAB) preconditioned by means of Schwarz additive scalable preconditioners. The properties of the new method are verified carrying out several numerical tests. At first, some elliptic, parabolic and convective-diffusive problems are solved and discussed, then the results of some time dependent and stationary 2D Navier-Stokes problems (in particular the well known benchmark problem of the natural convection in a square cavity) are discussed and compared to those found in the literature. Another, potentially very important application of the numerical tools developed, regards the solution of 1D Shallow-Water equations. In fact the use of the fractional steps scheme for advancing in time and the finite elements (of different polynomial degrees) for the spatial approximation, makes the above mentioned approach computationally profitable and convenient for real applications. The efficiency and accuracy of the numerical model have been checked by solving a theoretical test. Finally, a brief description of the software suitably developed and used in the tests conclude the thesis

Coupling different discretizations for fluid structure interaction in a monolithic approach

In this thesis we present a monolithic coupling approach for the simulation of phenomena involving interacting fluid and structure using different discretizations for the subproblems. For many applications in fluid dynamics, the Finite Volume method is the first choice in simulation science. Likewise, for the simulation of structural mechanics the Finite Element method is one of the most, if not the most, popular discretization method. However, despite the advantages of these discretizations in their respective application domains, monolithic coupling schemes have so far been restricted to a single discretization for both subproblems. We present a fluid structure coupling scheme based on a mixed Finite Volume/Finite Element method that combines the benefits of these discretizations. An important challenge in coupling fluid and structure is the transfer of forces and velocities at the fluidstructure interface in a stable and efficient way. In our approach this is achieved by means of a fully implicit formulation, i.e., the transfer of forces and displacements is carried out in a common set of equations for fluid and structure. We assemble the two different discretizations for the fluid and structure subproblems as well as the coupling conditions for forces and displacements into a single large algebraic system. Since we simulate real world problems, as a consequence of the complexity of the considered geometries, we end up with algebraic systems with a large number of degrees of freedom. This necessitates the use of parallel solution techniques. Our work covers the design and implementation of the proposed heterogeneous monolithic coupling approach as well as the efficient solution of the arising large nonlinear systems on distributed memory supercomputers. We apply Newton’s method to linearize the fully implicit coupled nonlinear fluid structure interaction problem. The resulting linear system is solved with a Krylov subspace correction method. For the preconditioning of the iterative solver we propose the use of multilevel methods. Specifically, we study a multigrid as well as a two-level restricted additive Schwarz method. We illustrate the performance of our method on a benchmark example and compare the afore mentioned different preconditioning strategies for the parallel solution of the monolithic coupled system

An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalized decomposition

A non-intrusive proper generalized decomposition (PGD) strategy, coupled with an overlapping domain decomposition (DD) method, is proposed to efficiently construct surrogate models of parametric linear elliptic problems. A parametric multi-domain formulation is presented, with local subproblems featuring arbitrary Dirichlet interface conditions represented through the traces of the finite element functions used for spatial discretization at the subdomain level, with no need for additional auxiliary basis functions. The linearity of the operator is exploited to devise low-dimensional problems with only few active boundary parameters. An overlapping Schwarz method is used to glue the local surrogate models, solving a linear system for the nodal values of the parametric solution at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the overlapping region. The proposed DD-PGD methodology relies on a fully algebraic formulation allowing for real-time computation based on the efficient interpolation of the local surrogate models in the parametric space, with no additional problems to be solved during the execution of the Schwarz algorithm. Numerical results for parametric diffusion and convection-diffusion problems are presented to showcase the accuracy of the DD-PGD approach, its robustness in different regimes and its superior performance with respect to standard high-fidelity DD methods

Uniform subspace correction preconditioners for discontinuous Galerkin methods with hphp-refinement

In this paper, we develop subspace correction preconditioners for discontinuous Galerkin (DG) discretizations of elliptic problems with $hp$-refinement. These preconditioners are based on the decomposition of the DG finite element space into a conforming subspace, and a set of small nonconforming edge spaces. The conforming subspace is preconditioned using a matrix-free low-order refined technique, which in this work we extend to the $hp$-refinement context using a variational restriction approach. The condition number of the resulting linear system is independent of the granularity of the mesh $h$, and the degree of polynomial approximation $p$. The method is amenable to use with meshes of any degree of irregularity and arbitrary distribution of polynomial degrees. Numerical examples are shown on several test cases involving adaptively and randomly refined meshes, using both the symmetric interior penalty method and the second method of Bassi and Rebay (BR2).Comment: 24 pages, 9 figure