In this article we consider the development of discontinuous Galerkin finite element methods for the numerical approximation of the compressible Navier-Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the interior penalty method, originally developed for the numerical approximation of linear self-adjoint second-order elliptic partial differential equations. In order to solve the resulting system of nonlinear equations, we exploit a (damped) Newton-GMRES algorithm. Numerical experiments demonstrating the practical performance of the proposed discontinuous Galerkin method with higher-order polynomials are presented.

Designing effective high-order adaptive methods for solving stationary reaction-diffusion equations in three dimensions requires the selection of a finite element basis, a posteriori error estimator and refinement strategy. Estimator accuracy may depend on the basis chosen, which in turn, may lead to unreliability or inefficiency via under- or over-refinement, respectively. The basis may also have an impact on the size and condition of the matrices that arise from discretization, and thus, on algorithm effectiveness. Herein, the interaction between these three components is studied in the context of an $h$-refinement procedure. The effects of these choices on the robustness and efficiency of the algorithm are examined for several linear and nonlinear problems. The results demonstrate that popular choices such as the tensor-product basis or the modified Szabό-Babuška basis have significant shortcomings but that promising alternatives exist.

We consider the $hp$-version interior penalty discontinuous Galerkin finite element method ($hp$-DGFEM) for linear second-order elliptic reaction-diffusion-advection equations with mixed Dirichlet and Neumann boundary conditions. Our main concern is the extension of the error analysis of the $hp$-DGFEM to the case when anisotropic (shape-irregular) elements and anisotropic polynomial degrees are used. For this purpose, extensions of well known approximation theory results are derived. In particular, new error bounds for the approximation error of the $L^2$-and $H^1$-projection operators are presented, as well as generalizations of existing inverse inequalities to the anisotropic setting. Equipped with these theoretical developments, we derive general error bounds for the $hp$-DGFEM on anisotropic meshes, and anisotropic polynomial degrees. Moreover, an improved choice for the (user-defined) discontinuity-penalisation parameter of the method is proposed, which takes into account the anisotropy of the mesh. These results collapse to previously known ones when applied to problems on shape-regular elements. The theoretical findings are justified by numerical experiments, indicating that the use of anisotropic elements, together with our newly suggested choice of the discontinuity-penalisation parameter, improves the stability, the accuracy and the efficiency of the method.

A new method for the solution of Burgers' equation is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details. The marker method is applicable to a general class of nonlinear dispersive partial differential equations.

We study computational methods for linear, degenerate advection-diffusion equations leading to coupled hyperbolic-parabolic problems. A multi-algorithmic approach is proposed in which a different approximation method is used locally depending on the mathematical nature of the problem. Our analysis focuses on stability and a priori error estimates of coupled continuous and discontinuous Galerkin methods, achieving a global $h^{p+\frac{1}{2}}$ estimate. Both the mathematical analysis and the numerical results demonstrate that careful consideration is necessary when defining appropriate interface conditions between the hyperbolic and parabolic regions.

Numerical investigation of BiCG and GMRES methods for solving non-symmetric linear equation systems with dominant skew-symmetric part has been presented. Numerical experiments were carried out for the linear system arising from a 5-point central difference approximation of the two dimensional convection-diffusion problem with different velocity coefficients and small parameter at the higher derivative. Behavior of BiCG and GMRES(10) has been compared for such kind of systems.