Abstract
We consider a norm-preconditioning approach for the solution of discontinuous Galerkin finite element discretizations of second order partial differential equations with a nonnegative characteristic form. Our solution method is a norm-preconditioned three-term GMRES routine. We find that for symmetric positive-definite diffusivity tensors the convergence of our solver is independent of discretization, while for the semidefinite case both theory and experiment indicate dependence on both h and p. Numerical results are included to illustrate performance on several test cases.
Highlights
Recent years have seen an increasing interest in a class of non-conforming finite element approximations of elliptic boundary-value problems with hyperbolic nature, usually referred to as discontinuous Galerkin finite element methods
discontinuous Galerkin finite element methods (DGFEMs) admit good stability properties, they offer flexibility in the mesh design and in the imposition of boundary conditions (Dirichlet boundary conditions are weakly imposed), and they are increasingly popular in the context of hp-adaptive algorithms
Existing approaches to solving systems arising in DGFEMs include domain decomposition, either non-overlapping [12], [13] or overlapping [23] and multigrid [19], [5]
Summary
Recent years have seen an increasing interest in a class of non-conforming finite element approximations of elliptic boundary-value problems with hyperbolic nature, usually referred to as discontinuous Galerkin finite element methods. The concept of norm-equivalence was formally introduced in [11] in the context of preconditioning for standard finite element discretisation of elliptic problems. ∂−κ := {x ∈ ∂κ : b(x) · n(x) < 0}, ∂+κ := {x ∈ ∂κ : b(x) · n(x) > 0}, where n(·) denotes the unit outward normal vector function associated with the element κ; we call these the inflow and outflow parts of ∂κ respectively.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
