Volume 11, Issue 4

A transmission eigenvalue problem on two disjoint intervals has been investigated. A distribution of eigenvalues has been obtained. A corresponding difference scheme is proposed and tested with a few numerical examples.

We consider an example of a boundary value problem on an interval where the solution can show strong oscillations. In order to solve such a problem numerically, standard methods require meshes that resolve these oscillations and will thus need a prohibitively large number of unknowns. In our approach we use special problem dependent basis functions in the finite element method and provide an analysis for a priori and a posteriori bounds. In this way we can construct an efficient approximation method for the solution of such boundary value problems.

Recent results on convergence of fully discrete approximations combining the Galerkin method with the explicit-implicit Euler scheme are extended to strong convergence under additional monotonicity assumptions. It is shown that these abstract results, formulated in the setting of evolution equations, apply, for example, to the partial differential equation for vibrating membrane with nonlinear damping and to another partial differential equation that is similar to one of the equations used to describe martensitic transformations in shape-memory alloys. Numerical experiments are performed for the vibrating membrane equation with nonlinear damping which support the convergence results.

We provide a MATLAB package p1afem for an adaptive P1-finite element method (AFEM). This includes functions for the assembly of the data, different error estimators, and an indicator-based adaptive meshrefining algorithm. Throughout, the focus is on an efficient realization by use of MATLAB built-in functions and vectorization. Numerical experiments underline the efficiency of the code which is observed to be of almost linear complexity with respect to the runtime. Although the scope of this paper is on AFEM, the general ideas can be understood as a guideline for writing efficient MATLAB code.

A functional-discrete method for solving the Cauchy problem for first-order ordinary differential equations (ODEs) is proposed. Generally speaking, this method (FD-method) is similar to the Adomian Decomposition Method (ADM). However, we have shown that for some problems the FD-method is convergent whereas the ADM is divergent. The results presented in this paper can easily be generalized to the case of systems of ODEs.

We continue our study on variable order Arnold-Falk-Winther elements reported in [W. Qiu, L. Demkowicz, Mixed hp-finite element method for linear elasticity with weakly imposed symmetry. Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 3682-–3701] and [W. Qiu, L. Demkowicz, Mixed hp-finite element method for linear elasticity with weakly imposed symmetry: stability analysis. SIAM J. Numer. Anal., 49 (2011), pp. 619-–641] for 2D elasticity in context of parametric curvilinear elements. We present an asymptotic h-stability result.

For trigonometric polynomial approximation on a circle, the century-old de la Vallée-Poussin construction has attractive features: it exhibits uniform convergence for all continuous functions as the degree of the trigonometric polynomial goes to infinity, yet it also has arbitrarily fast convergence for sufficiently smooth functions. This paper presents an explicit generalization of the de la Vallée-Poussin construction to higher dimensional spheres S^d ≤ R^{d+1}. The generalization replaces the C^∞ filter introduced by Rustamov by a piecewise polynomial of minimal degree. For the case of the circle the filter is piecewise linear, and recovers the de la Vallée-Poussin construction, while for the general sphere S^d the filter is a piecewise polynomial of degree d and smoothness C^{d−1}. In all cases the approximation converges uniformly for all continuous functions, and has arbitrarily fast convergence for smooth functions.