HP-finite element approximations on non-matching grids for partial differential equations with non-negative characteristic form

Andrea Toselli
2003 Mathematical Modelling and Numerical Analysis  
We propose and analyze a domain decomposition method on non-matching grids for partial differential equations with non-negative characteristic form. No weak or strong continuity of the finite element functions, their normal derivatives, or linear combinations of the two is imposed across the boundaries of the subdomains. Instead, we employ suitable bilinear forms defined on the common interfaces, typical of discontinuous Galerkin approximations. We prove an error bound which is optimal with
more » ... is optimal with respect to the mesh-size and suboptimal with respect to the polynomial degree. Our analysis is valid for arbitrary shape-regular meshes and arbitrary partitions into subdomains. Our method can be applied to advective, diffusive, and mixed-type equations, as well, and is well-suited for problems coupling hyperbolic and elliptic equations. We present some two-dimensional numerical results that support our analysis for the case of linear finite elements. Mathematics Subject Classification. 65N12, 65N30, 65N35, 65N55. where Ω ⊂ R n , n = 2, 3, is a bounded, connected, Lipschitz polygon or polyhedron, with outward unit normal n. Here, A is a symmetric, positive-semidefinite matrix in Ω, b a given velocity field, c a non-negative reaction coefficient that may arise from a finite difference discretization of a time derivative, and f is a source term. In the next section, we make further hypotheses on L and we introduce appropriate boundary conditions. The aim of this paper is to construct and analyze an hp-finite element method for problem (1) on nonmatching grids. We propose an approach which is typical of discontinuous Galerkin (DG) methods, where finite element spaces consisting of discontinuous functions are considered. In particular, in a DG approach no
doi:10.1051/m2an:2003018 fatcat:tfktsslhqzb7ppxuw7d5ikipbu