Optimal Error Estimates for the hp–Version Interior Penalty Discontinuous Galerkin Finite Element Method

We consider the hp-version interior penalty discontinuous Galerkin finite element method (hp-DGFEM) for second-order linear reaction-diffusion equations. To the best of our knowledge, the sharpest known error bounds for the hp-DGFEM are due to Riviere, Wheeler and Girault [9] and due to Houston, Schwab and Süli [6] which are optimal with respect to the meshsize h but suboptimal with respect to the polynomial degree p by half an order of p. We present improved error bounds in the energy norm, by introducing a new function space framework. More specifically, assuming that the solutions belong element-wise to an augmented Sobolev space, we deduce hp-optimal error bounds

A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems

We consider a variant of the hp-version interior penalty discontinuous Galerkin finite element method (IP-DGFEM) for second order problems of degenerate type. We do not assume uniform ellipticity of the diffusion tensor. Moreover, diffusion tensors or arbitrary form are covered in the theory presented. A new, refined recipe for the choice of the discontinuity-penalisation parameter (that is present in the formlation of the IP-DGFEM) is given. Making use of the recently introduced augmented Sobolev space framework, we prove an hp-optimal error bound in the energy norm and an h-optimal and slightly p-suboptimal (by only half an order of p) bound in the L2 norm, provided that the solution belongs to an augmented Sobolev space

hp-DGFEM on Shape-Irregular Meshes: Reaction-Diffusion Problems

We consider the hp-version of the discontinuous Galerkin finite element method (DGFEM) for second-order elliptic reaction-diffusion equations with mixed Dirichlet and Neumann boundary conditions. For simplicity of the presentation, we only consider boundary-value problems defined on an axiparallel polygonal domain whose solutions are approximated on subdivisions consisting of axiparallel elements. Our main concern is the generalisation of the error analysis of the hp-DGFEM for the case when shape-irregular (anisotropic) meshes and anisotropic polynomial degrees for the element basis functions are used. We shall present a general framework for deriving error bounds for the approximation error and we shall consider two important special cases. In the first of these we derive an error bound that is simultaneously optimal in h and p, for shape-regular elements and isotropic polynomial degrees, provided that the solution belongs to a certain anisotropic Sobolev space. The second result deals with the case where we have a uniform polynomial degree in every space direction and a shape-irregular mesh. Again we derive an error bound that is optimal both in h and in p. For element-wise analytic solutions the method exhibits exponential rates of convergence under p-refinement, in both cases considered. Finally, numerical experiments using shape-regular and shape-irregular elements are presented

Adaptive discontinuous Galerkin approximations to fourth order parabolic problems

An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in space for linear parabolic fourth order problems is presented. The a posteriori analysis is performed for convex domains in two and three space dimensions for local spatial polynomial degrees $r\ge 2$. The a posteriori estimates are then used within an adaptive algorithm, highlighting their relevance in practical computations, which results into substantial reduction of computational effort

Discontinuous Galerkin Methods for Mass Transfer through Semi-Permeable Membranes

A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multi-compartment partial differential equation (PDE) models, interconnected with interface conditions, is presented and analysed. The study of interface problems is motivated by models of mass transfer of solutes through semi-permeable membranes. More specifically, a model problem consisting of a system of semilinear parabolic advection-diffusion-reaction partial differential equations in each compartment, equipped with respective initial and boundary conditions, is considered. Nonlinear interface conditions modelling selective permeability, congestion and partial reflection are applied to the compartment interfaces. An interior penalty dG method is presented for this problem and it is analysed in the space-discrete setting. The a priori analysis shows that the method yields optimal a priori bounds, provided the exact solution is sufficiently smooth. Numerical experiments indicate agreement with the theoretical bounds and highlight the stability of the numerical method in the advection-dominated regime