Paul Houston

Discontinuous hp -Finite Element Methods for Advection-Diffusion-Reaction Problems

We consider the hp-version of the discontinuous Galerkin finite element method (DGFEM) for second-order partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic equations, advection-reaction equations, as well as problems of mixed hyperbolic-elliptic-parabolic type. Our main concern is the error analysis of the method in the absence of streamline-diffusion stabilization. In the hyperbolic case, an hp-optimal error bound is derived; here, we consider only advection-reaction problems which satisfy a certain (standard) positivity condition. In the self-adjoint elliptic case, an error bound that is h-optimal and p-suboptimal by

ENERGY NORM A POSTERIORI ERROR ESTIMATION OF hp-ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC PROBLEMS

\frac{1}{2}

Adaptive Discontinuous Galerkin Finite Element Methods for Nonlinear Hyperbolic Conservation Laws

a power of p is obtained. These estimates are then combined to deduce an error bound in the general case. For elementwise analytic solutions the method exhibits exponential rates of convergence under p-refinement. The theoretical results are illustrated by numerical experiments.

hp -Adaptive Discontinuous Galerkin Finite Element Methods for First-Order Hyperbolic Problems

In this paper, we develop the a posteriori error estimation of hp-version interior penalty discontinuous Galerkin discretizations of elliptic boundary-value problems. Computable upper and lower bounds on the error measured in terms of a natural (mesh-dependent) energy norm are derived. The bounds are explicit in the local mesh sizes and approximation orders. A series of numerical experiments illustrate the performance of the proposed estimators within an automatic hp-adaptive refinement procedure.

Stabilized hp -Finite Element Methods for First-Order Hyperbolic Problems

We consider the a posteriori error analysis and adaptive mesh design for discontinuous Galerkin finite element approximations to systems of nonlinear hyperbolic conservation laws. In particular, we discuss the question of error estimation for general target functionals of the solution; typical examples include the outflow flux, local average and pointwise value, as well as the lift and drag coefficients of a body immersed in an inviscid fluid. By employing a duality argument, we derive so-called weighted or Type I a posteriori error bounds; these error estimates include the product of the finite element residuals with local weighting terms involving the solution of a certain dual or adjoint problem that must be numerically approximated. Based on the resulting approximate Type I bound, we design and implement an adaptive algorithm that produces meshes specifically tailored to the efficient computation of the given target functional of practical interest. The performance of the proposed adaptive strategy and the quality of the approximate Type I a posteriori error bound is illustrated by a series of numerical experiments. In particular, we demonstrate the superiority of this approach over standard mesh refinement algorithms which employ Type II error indicators.

Interior penalty method for the indefinite time-harmonic Maxwell equations

We consider the {a posteriori} error analysis of hp-discontinuous Galerkin finite element approximations to first-order hyperbolic problems. In particular, we discuss the question of error estimation for linear functionals, such as the outflow flux and the local average of the solution. Based on our {a posteriori} error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local polynomial-degree variation and local mesh subdivision. The theoretical results are illustrated by a series of numerical experiments.

A posteriori error analysis for stabilised finite element approximations of transport problems

We analyze the hp-version of the streamline-diffusion finite element method (SDFEM) and of the discontinuous Galerkin finite element method (DGFEM) for first-order linear hyperbolic problems. For both methods, we derive new error estimates on general finite element meshes which are sharp in the mesh-width h and in the spectral order p of the method, assuming that the stabilization parameter is O(h/p). For piecewise analytic solutions, exponential convergence is established on quadrilateral meshes. For the DGFEM we admit very general irregular meshes and for the SDFEM we allow meshes which contain hanging nodes. Numerical experiments confirm the theoretical results.

Mixed Discontinuous Galerkin Approximation of the Maxwell Operator

SummaryIn this paper, we introduce and analyze the interior penalty discontinuous Galerkin method for the numerical discretization of the indefinite time-harmonic Maxwell equations in the high-frequency regime. Based on suitable duality arguments, we derive a-priori error bounds in the energy norm and the L2-norm. In particular, the error in the energy norm is shown to converge with the optimal order (hmin{s,ℓ}) with respect to the mesh size h, the polynomial degree ℓ, and the regularity exponent s of the analytical solution. Under additional regularity assumptions, the L2-error is shown to converge with the optimal order (hℓ+1). The theoretical results are confirmed in a series of numerical experiments.

A posteriori error analysis for numerical approximations of Friedrichs systems

We develop the a posteriori error analysis of stabilised finite element approximations to linear transport problems via duality arguments. Two alternative dual problems are considered: one is based on the formal adjoint of the hyperbolic differential operator, the other on the transposition of the bilinear form for the stabilised finite element method. We show both analytically and through numerical experiments that the second approach is superior in the sense that it leads to sharper a posteriori error bounds and more economical adaptively refined meshes.

Adaptive Finite Element Approximation of Hyperbolic Problems

We introduce and analyze a discontinuous Galerkin discretization of the Maxwell operator in mixed form. Here, all the unknowns of the underlying system of partial differential equations are approximated by discontinuous finite element spaces of the same order. For piecewise constant coefficients, the method is shown to be stable and optimally convergent with respect to the mesh size. Numerical experiments highlighting the performance of the proposed method for problems with both smooth and singular analytical solutions are presented.