Band 12, Heft 3

We propose and analyze several two-level non-overlapping Schwarz methods for a preconditioned weakly over-penalized symmetric interior penalty (WOPSIP) discretization of a second order boundary value problem. We show that the preconditioners are scalable and that the condition number of the resulting preconditioned linear systems of equations is independent of the penalty parameter and is of order H/h , where H and h represent the mesh sizes of the coarse and fine partitions, respectively. Numerical experiments that illustrate the performance of the proposed two-level Schwarz methods are also presented.

In this paper, we introduce a numerical method for solving the space-time fractional telegraph equation. The numerical method is based on a quadrature formula approach and a stability condition for the numerical method is obtained. Two numerical examples are given and the stability regions are plotted.

Abstract — We have studied the stability of finite-difference schemes approximating initial-boundary value problem (IBVP) for multidimensional parabolic equations with a nonlinear source of a power type. We have obtained simple sufficient input data conditions, in which the solutions of differential and difference problems are globally bounded for all t. It is shown that if these conditions are not satisfied, then the solution can blow-up (go to infinity) in finite time. The lower bound of the blow-up time has been determined. The stability of the difference solution has been proven. In all cases, we used the method of energy inequalities based on the application of the Chaplygin comparison theorem for nonlinear ODEs, Bihari-type inequalities and their discrete analogs.

Simulation of flow phenomena in the ocean and in other large but relatively flat basins are typically based on the so-called primitive equations, which, among others, result from application of the hydrostatic approximation. The crucial premise for this approximation is the dominance of the hydrostatic balance over remaining vertical flow phenomena in large but flat domains, which leads to a decomposition of the three-dimensional (3D) pressure field into a hydrostatic part and an only two-dimensional (2D) hydrodynamic part. The former pressure can be obtained by solving Ordinary Differential Equations. The latter one is determined by a 2D elliptic problem which can be solved quite efficiently. The velocity field remains three dimensional. However, its vertical component can be eliminated from the dynamic system. In this work, we analyze such "2.5-dimensional" (2.5D) Stokes systems and formulate stabilized finite element schemes with equal-order interpolation. The absence of a discrete inf-sup condition is compensated by introducing additional terms into the discrete variational form. We show stability and give an a priori error estimate for several established stabilized equal-order schemes, as pressure-stabilized Petrov-Galerkin (PSPG), Galerkin least squares (GLS) and local projection schemes (LPS) which are extended here to the hydrostatic approximation. The basic assumption we need is a certain property of the underlying 3D mesh. We illustrate the order of convergence of the 2.5D problem by numerical examples and demonstrate the effect of the hydrostatic approximation in comparison to the full 3D problem.

The problem under consideration is the three-dimensional transmission problem for time harmonic acoustic waves for two homogeneous media. A simply-connected and bounded region with sufficiently smooth boundary is immersed in an infinite {medium. Each medium is c haracterized by the space independent wave number κ and the density μ.} The system of boundary integral equations is reviewed as well as an existence and uniqueness result. The system is approximated by the boundary element collocation method and consistency, stability, and convergence is proved. In addition, superconvergence is proved and numerical results illustrate the agreement with these theoretical results. No numerical results seem to be reported for this method yet.

Starting from suitable zero-relation, we derive higher-order iterative methods for the simultaneous inclusion of polynomial multiple zeros in circular complex interval arithmetic. The convergence rate is increased using a family of two-point methods of the fourth order for solving nonlinear equations as a predictor. The methods are more efficient compared to existing inclusion methods for multiple zeros, based on fixed point relations. Using the concept of the R -order of convergence of mutually dependent sequences, we present the convergence analysis of the total-step and the single-step methods. The proposed self-validated methods possess a great computational efficiency since the acceleration of the convergence rate from four to seven is achieved only by a few additional calculations. To demonstrate convergence behavior of the presented methods, two numerical examples are given.