Andrea Toselli

Mixed hp -DGFEM for Incompressible Flows

We consider several mixed discontinuous Galerkin approximations of the Stokes problem and propose an abstract framework for their analysis. Using this framework, we derive a priori error estimates for hp-approximations on tensor product meshes. We also prove a new stability estimate for the discrete divergence bilinear form.

An Overlapping Domain Decomposition Preconditioner for a Class of Discontinuous Galerkin Approximations of Advection-Diffusion Problems

We consider a scalar advection-diffusion problem and a recently proposed discontinuous Galerkin approximation, which employs discontinuous finite element spaces and suitable bilinear forms containing interface terms that ensure consistency. For the corresponding sparse, non-symmetric linear system, we propose and study an additive, two--level overlapping Schwarz preconditioner, consisting of a coarse problem on a coarse triangulation and local solvers associated to suitable problems defined on a family of subdomains. This is a generalization of the corresponding overlapping method for approximations on continuous finite element spaces. Related to the lack of continuity of our approximation spaces, some interesting new features arise in our generalization, which have no analog in the conforming case. We prove an upper bound for the number of iterations obtained by using this preconditioner with GMRES, which is independent of the number of degrees of freedom of the original problem and the number of subdomains. The performance of the method is illustrated by several numerical experiments for different test problems, using linear finite elements in two dimensions.

A FETI Preconditioner for Two Dimensional Edge Element Approximations of Maxwell's Equations on Nonmatching Grids

A class of FETI methods for the mortar approximation of a vector field problem in two dimensions is proposed. Edge element discretizations of lowest degree are considered. The method proposed can be employed with geometrically conforming and nonconforming partitions. Our numerical results show that its condition number increases only with the number of unknowns in each subdomain and is independent of the number of subdomains and the size of the problem.

A FETI Domain Decomposition Method for Edge Element Approximations in Two Dimensions with Discontinuous Coefficients

A class of finite element tearing and interconnecting (FETI) methods for the edge element approximation of vector field problems in two dimensions is introduced and analyzed. First, an abstract framework is presented for the analysis of a class of FETI methods where a natural coarse problem, associated with the substructures, is lacking. Then, a family of FETI methods for edge element approximations is proposed. It is shown that the condition number of the corresponding method is independent of the number of substructures and grows only polylogarithmically with the number of unknowns associated with individual substructures. The estimate is also independent of the jumps of both of the coefficients of the original problem. Numerical results validating our theoretical bounds are given. The method and its analysis can be easily generalized to Raviart--Thomas element approximations in two and three dimensions.

Stabilized hp-DGFEM for Incompressible Flow

We consider stabilized mixed hp-discontinuous Galerkin methods for the discretization of the Stokes problem in three-dimensional polyhedral domains. The methods are stabilized with a term penalizing the pressure jumps. For this approach it is shown that ℚk-ℚk and ℚk-ℚk-1 elements satisfy a generalized inf–sup condition on geometric edge and boundary layer meshes that are refined anisotropically and non quasi-uniformly towards faces, edges, and corners. The discrete inf–sup constant is proven to be independent of the aspect ratios of the anisotropic elements and to decrease as k-1/2 with the approximation order. We also show that the generalized inf–sup condition leads to a global stability result in a suitable energy norm.

Convergence of Some Two-Level Overlapping Domain Decomposition Preconditioners with Smoothed Aggregation Coarse Spaces

We study two-level overlapping preconditioners with smoothed aggregation coarse spaces for the solution of sparse linear systems arising from finite element discretizations of second order elliptic problems. Smoothed aggregation coarse spaces do not require a coarse triangulation. After aggregation of the fine mesh nodes, a suitable smoothing operator is applied to obtain a family of overlapping subdomains and a set of coarse basis functions. We consider a set of algebraic assumptions on the smoother, that ensure optimal bounds for the condition number of the resulting preconditioned system. These assumptions only involve geometrical quantities associated to the sub domains, namely the diameter of the sub domains and the overlap. We first prove an upper bound for the condition number, which depends quadratically on the relative overlap. If additional assumptions on the coarse basis functions hold, a linear bound can be found. Finally, the performance of the preconditioners obtained by different smoothing procedures is illustrated by numerical experiments for linear finite elements in two dimensions.

A numerical study on Neumann–Neumann and FETI methods for hp approximations on geometrically refined boundary layer meshes in two dimensions

Abstract We present extensive numerical tests showing the performance and robustness of certain Balancing Neumann–Neumann and one-level FETI methods for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in two dimensions. The numerical results confirm the theoretical results derived in [A. Toselli, X. Vasseur, Technical Report 02-15, Seminar fur Angewandte Mathematik, ETH, Zurich, September 2002]: the condition numbers are independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients. In addition, they only grow polylogarithmically with the polynomial degree, as in the case of p approximations on shape-regular meshes.

Mixed HP-finite element approximations on geometric edge and boundary layer meshes in three dimensions

SummaryIn this paper, we consider the Stokes problem in a three-dimensional polyhedral domain discretized with hp finite elements of type Qk for the velocity and Qk-2 for the pressure, defined on hexahedral meshes anisotropically and non quasi-uniformly refined towards faces, edges, and corners. The inf-sup constant of the discretized problem is independent of arbitrarily large aspect ratios. Our work generalizes a recent result for two-dimensional problems in [10, 11].

Robust and efficient FETI domain decomposition algorithms for edge element approximations

Purpose – A family of preconditioned dual‐primal FETI iterative algorithms for the solution of algebraic systems arising from edge element approximations in two dimensions is presented.Design/methodology/approach – The primal constraints, which determine the size of the coarse problem to be solved at each iteration step, are here suitable averages over subdomain edges. The condition number of the corresponding methods is independent of the number of subdomains and possibly large jumps of the coefficients.Findings – For h finite elements, it grows only polylogarithmically with the number of unknowns associated with individual substructures, while for hp approximations on geometrically refined meshes, it is independent of arbitrarily large aspect ratios.Originality/value – Proposes an algorithm with a rate of convergence that is independent of possibly large jumps of the coefficients and mesh aspect ratios.

Mixed hp-finite element approximations on geometric boundary layer meshes in R3

Abstract In this paper, we consider the Stokes problem in a three-dimensional polyhedral domain discretized with hp finite elements of type Q k for the velocity and Q k−2 for the pressure, defined on meshes anisotropically and non-quasi-uniformly refined towards faces, edges, and corners. The inf-sup constant of the discretized problem is independent of arbitrarily large aspect ratios and exhibits the same dependence on k as in in the case of isotropically refined meshes. Our work generalizes a recent result for two–dimensional problems in [4,5].