Band 14, Heft 3

Abstract. In this work analyses are carried out to verify the well-posedness of two three-field formulations for linear elasticity that have been shown to work well in practice, but for which analyses are absent. Both are formulated in three dimensions on tetrahedral elements. The first has as primary unknowns the displacement, pressure, and dilatation, and the second is a formulation with primary unknowns the displacement, pressure and enhanced strain, with the space of enhanced strains spanned by functions that are bubble functions on a surface of the element. It is shown that the first-mentioned formulation is a special case of a general three-field formulation, for which well-posedness conditions have been established. These conditions are used here to show that the pressure-dilatation-displacement formulation is uniformly convergent in the incompressible limit. The pressure-enhanced strain-displacement formulation is not amenable to such an approach; instead, it is reformulated in the form of a standard discrete saddle point problem, and the uniform convergence of the formulation established directly by verifying the ellipticity and inf-sup conditions.

Abstract. With newest vertex bisection, there is no uniform bound on the number of n -simplices that need to be refined to arrive at the smallest conforming refinement 𝒯'$\mathcal {T}^{\prime }$ of a conforming partition 𝒯$\mathcal {T}$ in which one simplex has been bisected. In this note, we show that the difference in levels between any T'∈𝒯'$T^{\prime } \in \mathcal {T}^{\prime }$ and its ancestor T∈𝒯$T \in \mathcal {T}$ is uniformly bounded. This result has been used in Lemma 4.2 of [SIAM J. Numer. Anal. 51 (2013), 2935–2955] by Carstensen and the first two authors.

Abstract. In this paper, a brief review of tailored finite point methods (TFPM) is given. The TFPM is a new approach to construct the numerical solutions of partial differential equations. The TFPM has been tailored based on the local properties of the solution for each given problem. Especially, the TFPM is very efficient for solutions which are not smooth enough, e.g., for solutions possessing boundary/interior layers or solutions being highly oscillated. Recently, the TFPM has been applied to singular perturbation problems, the Helmholtz equation with high wave numbers, the first-order wave equation in high frequency cases, transport equations with interface, second-order elliptic equations with rough or highly oscillatory coefficients, etc.

Abstract. We consider direct and inverse scattering problems for a general nonlinear Schrödinger equation. The direct problem is solved by the method of integral equations. The inverse problem is solved by the method of Born approximation. The computation of the Born approximation is carried out using regularization methods. Numerical examples are given to illustrate the effectiveness of these methods.

Abstract. We construct an usual linearized difference scheme for initial boundary value problems (IBVP) for one-dimensional quasilinear parabolic equations with generalized solutions. The uniform parabolicity condition 0<k1≤k(u)≤k2$0&lt;k_1\le k(u) \le k_2$ is assumed to be fulfilled for the sign alternating solution u(x,t)∈D¯(u)$u(x,t) \in \bar{D}(u)$ only in the domain of exact solution values (unbounded non-linearity). On the basis of new corollaries of the maximum principle, we establish not only two-sided estimates for the approximate solution y but its belonging to the domain of exact solution values. We assume that the solution is continuous and its first derivative ∂u∂x$\frac{\partial u}{\partial x}$ has discontinuity of the first kind in the neighborhood of the finite number of discontinuity lines. An existence of time derivative in any sense is not assumed. We prove convergence of the approximate solution to the generalized solution of the differential problem in the grid norm L 2 .

Abstract. We derive computable bounds of deviations from the exact solution of the stationary Oseen problem. They are applied to approximations generated by the Uzawa iteration method. Also, we derive an advanced form of the estimate, which takes into account approximation errors arising due to discretization of the boundary value problem, generated by the main step of the Uzawa method. Numerical tests confirm our theoretical results and show practical applicability of the estimates.

Abstract. We propose a method to calculate all zeros of an analytic function in a given rectangle. The main idea of our method is to construct a covering of the initial rectangle by subsets where either there are no zeros or the zero uniqueness test based on the Kantorovich theorem is satisfied. The algorithm for the construction of such covering is presented. The implementation of the method is shown on different examples.

Abstract. Let ( M , g ) be a smooth and complete surface, Ω⊂M$\Omega \subset M$ be a domain in M , and Δg$\Delta _g$ be the Laplace operator on M . The spectrum of the Dirichlet–Laplace operator on Ω is a sequence 0<λ1(Ω)≤λ2(Ω)≤⋯↗∞$0 &lt; \lambda _1(\Omega ) \le \lambda _2(\Omega ) \le \cdots \nearrow \infty $. A classical question is to ask what is the domain Ω*$\Omega ^*$ which minimizes λm(Ω)$\lambda _m(\Omega )$ among all domains of a given area, and what is the value of the corresponding λm(Ωm*)$\lambda _m(\Omega _m^*)$. The aim of this article is to present a numerical algorithm using shape optimization and based on the finite element method to find an approximation of a candidate for Ωm*$\Omega _m^*$. Some verifications with existing numerical results are carried out for the first eigenvalues of domains in ℝ 2 . Furthermore, some investigations are presented in the two-dimensional sphere to illustrate the case of the positive curvature, in hyperbolic space for the negative curvature and in a hyperboloid for a non-constant curvature.

Abstract. A boundary element method with higher order isoparametric elements allows for the simulation of scattering problems given by the electric field integral equation. This paper describes the implementation and provides striking numerical evidence of the higher order convergence rates for the approximation of the electric surface current. One key issue is the necessity of higher order geometrical approximation which leads to an extreme efficiency of some mesh-dependent variable-order approximation.