Previous theoretical and computational investigations have shown high efficiency of the local Green's function method for the numerical solution of singularly perturbed problems with sharp boundary layers. However, in several space variables those functions, used as projectors in the Petrov-Galerkin scheme, cannot be derived in a closed analytical form. This is an obstacle for the application of the method when applied to multi-dimensional problems. The present work proposes a semi-analytical approach to calculate the local Green's function, which opens a way to effective practical application of the method. Besides very accurate approximation, the matrix stencils obtained with these functions allow the use of fast and stable iterative solution of the large sparse algebraic systems that arise from the grid-discretization. The advantages of the method are illustrated by numerical examples.

We propose a variant of variable preconditioning for Generalized Conjugate Residual (CCR)-like methods. The preconditioning is carried out by roughly solving $Az = v$ by an iterative method to a certain degree of accuracy instead of computing $Kz = v$ in a conventional preconditioned algorithm. In our proposal, the number of iterations required for computing $Az = v$ is changed at each iteration by establishing a stopping criterion. This enables the use of a stationary iterative method when applying different preconditioners. The proposed procedure is incorporated into GCR, and the mathematical convergence is proved. In numerical experiments, we employ the Successive Over-Relaxation (SOR) method for computing $Az = v$, and we demonstrate that GCR with the variable preconditioning using SOR is faster and more robust than GCR with an incomplete LU preconditioning, and the FGMRES and GMRESR methods with the variable preconditioning using the Generalized Minimal Residual (GMRES) method. Moreover, we confirm that different preconditioners are applied at each iteration.

In this paper we demonstrate the performance of a slope limiting procedure combined with a discontinuous Galerkin (DG) finite element solver for 2D compressible Euler equations. The slope limiter can be categorized into van Albada type and is differentiable. This slope limiter is modified from a similar limiter used in finite volume solvers to suit the needs of the DC solver. The gradient in an element is limited using the weighted average of the face gradients. The face gradients are obtained from the area-weighted average of the gradient on both sides of the faces. The slope limiting process is very suitable for meshes discretized by triangle elements. The HLLC (Harten, Lax and van Leer) or the local Lax-Friedrich (LLF) flux functions is used to compute the interface fluxes in the DG formulation. The second order TVD Runge-Kutta scheme is employed for the time integration. The numerical examples including transonic, supersonic and hypersonic flows show that the current slope limiting process together with the DG solver is able to remove overshoots and undershoots around high gradient regions while preserving the high accuracy of the DC method. The convergence histories of all examples demonstrate that the limiting process does not stall convergence to steady state as many other slope limiters do.

This paper deals with an incompressible viscous flow problem, where the Navier-Stokes equations are coupled with a nonlinear heat equation. Existence and uniqueness results are established. Next, a finite element approximation of the problem is presented and analyzed. Error estimates are obtained and a posteriori error estimate is given.

Several convergence and stability issues of the balanced implicit methods (BIMs) for systems of real-valued ordinary stochastic differential equations are thoroughly discussed. These methods are linear-implicit ones, hence easily implementable and computationally more efficient than commonly known nonlinear-implicit methods. In particular, we relax the so far known convergence condition on its weight matrices $c^j$. The presented convergence proofs extend to the case of nonrandom variable step sizes and show a dependence on certain Lyapunov-functionals $V$ : $\rm{IR}^d \rightarrow \rm{IR}_+^1$. The proof of $L^2$-convergence with global rate 0.5 is based on the stochastic Kantorovich-Lax-Richtmeyer principle proved by the author (2002). Eventually, $p$-th mean stability and almost sure stability results for martingale-type test equations document some advantage of BlMs. The problem of weak convergence with respect to the test class $C_{b(\kappa)}^2 (\rm{IR}^d, \rm{IR}^1)$ and with global rate 1.0 is tackled too.

In this paper, we consider mixed finite elements discretizations of a class of Quasi-Newtonian Stokes flow problem. Unified a posteriori error estimator for conforming, nonconforming, with or without stabilization is obtained. We prove, without Helmholtz decomposition of the error, nor regularity and saturation assumptions, the reliability and the efficiency of our estimator.