Volume 14, Issue 2

Abstract. A functional type a posteriori error estimator for the finite element discretization of the stationary reaction-convection-diffusion equation is derived. In case of dominant convection, the solution for this class of problems typically exhibits boundary layers and shock-front like areas with steep gradients. This renders the accurate numerical solution very demanding and appropriate techniques for the adaptive resolution of regions with large approximation errors are crucial. Functional error estimators as derived here contain no mesh-dependent constants and provide guaranteed error bounds for any conforming approximation. To evaluate the error estimator, a minimization problem is solved which does not require any Galerkin orthogonality or any specific properties of the employed approximation space. Based on a set of numerical examples, we assess the performance of the new estimator. It is observed that it exhibits a good efficiency also with convection-dominated problem settings.

Abstract. Initial boundary value problems of the Dirichlet type for quasilinear functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both classes of the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. Error estimates for both methods are presented. Interpolating operators corresponding to functional variables are constructed.

Abstract. We develop a new central-upwind scheme for a one-dimensional Savage–Hutter type model of submarine landslides and generated tsunami waves. Our scheme exactly preserves physically relevant steady-states, preserves positivity of water depth, is insensitive to choice of discretization of nonconservative products, and properly incorporates friction inherent in the model. We apply our scheme to a variety of test problems and the numerical results clearly demonstrate a high accuracy and robustness of the proposed method.

Abstract. In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, by combining three substeps of the theta time-stepping scheme, the fractional step theta time-stepping scheme is derived. It possesses highly desired stability and numerical dissipation properties and is second order accurate. The derived error estimator is based on a Petrov–Galerkin formulation that is up to a numerical quadrature error equivalent to the theta time-stepping scheme. The error estimator is assembled as one weighted residual term given by the dual weighted residual method and one additional residual estimating the Galerkin error between time-stepping scheme and Petrov–Galerkin formulation.

Abstract. This paper is concerned with the stability of Petrov–Galerkin discretizations with application to parabolic evolution problems in space-time weak form. We will prove that the discrete inf-sup condition for an a priori fixed Petrov–Galerkin discretization is satisfied uniformly under standard approximation and smoothness conditions without any further coupling between the discrete trial and test spaces for sufficiently regular operators. It turns out that one needs to choose different discretization levels for the trial and test spaces in order to obtain a positive lower bound for the discrete inf-sup condition which is independent of the discretization levels. In particular, we state the required number of extra layers in order to guarantee uniform boundedness of the discrete inf-sup constants explicitly. This general result will be applied to the space-time weak formulation of parabolic evolution problems as an important model example. In this regard, we consider suitable hierarchical families of discrete spaces. The results apply, e.g., for finite element discretizations as well as for wavelet discretizations. Due to the Riesz basis property, wavelet discretizations allow for optimal preconditioning independently of the grid spacing. Moreover, our predictions on the stability, especially in view of the dependence on the refinement levels w.r.t. the test and trial spaces, are underlined by numerical results. Furthermore, it can be observed that choosing the same discretization levels would, indeed, lead to stability problems.

Abstract. We present two novel coarse spaces ( H 1 - and H(curl)$H(\operatorname{curl})$-conforming) based on element agglomeration on unstructured tetrahedral meshes. Each H 1 -conforming coarse basis function is continuous and piecewise-linear with respect to an original tetrahedral mesh. The H(curl)$H(\operatorname{curl})$-conforming coarse space is a subspace of the lowest order Nédélec space of the first type. The H 1 -conforming coarse space exactly interpolates affine functions on each agglomerate. The H(curl)$H(\operatorname{curl})$-conforming coarse space exactly interpolates vector constants on each agglomerate. Combined with the H(div)$H(\operatorname{div})$- and L 2 -conforming spaces developed previously in [Numer. Linear Algebra Appl. 19 (2012), 414–426], the newly constructed coarse spaces form a sequence (with respect to exterior derivatives) which is exact as long as the underlying sequence of fine-grid spaces is exact. The constructed coarse spaces inherit the approximation and stability properties of the underlying fine-grid spaces supported by our numerical experiments. The new coarse spaces, in addition to multigrid, can be used for upscaling of broad range of PDEs involving curl$\operatorname{curl}$, div$\operatorname{div}$ and grad$\operatorname{grad}$ differential operators.