Volume 13, Issue 3

Abstract. We study spatially semidiscrete and fully discrete finite volume element methods for the homogeneous heat equation with homogeneous Dirichlet boundary conditions and derive error estimates for smooth and nonsmooth initial data. We show that the results of our earlier work [Math. Comp. 81 (2012), 1–20] for the lumped mass method carry over to the present situation. In particular, in order for error estimates for initial data only in L 2 to be of optimal second order for positive time, a special condition is required, which is satisfied for symmetric triangulations. Without any such condition, only first order convergence can be shown, which is illustrated by a counterexample. Improvements hold for triangulations that are almost symmetric and piecewise almost symmetric.

Abstract. We study the convergence of finite difference schemes for approximating elliptic equations of second order with discontinuous coefficients. Two of these finite difference schemes arise from the discretization by the finite element method using bilinear shape functions. We prove an convergence for the gradient, if the solution is locally in H 3 . Thus, in contrast to the first order convergence for the gradient obtained by the finite element theory we show that the gradient is superclose. From the Bramble–Hilbert Lemma we derive a higher order compact (HOC) difference scheme that gives an approximation error of order four for the gradient. A numerical example is given.

Abstract. We consider the extension of the p -robust equilibrated error estimator due to Braess, Pillwein and Schöberl to linear elasticity. We derive a formulation where the local mixed auxiliary problems do not require symmetry of the stresses. The resulting error estimator is p -robust, and the reliability estimate is also robust in the incompressible limit if quadratics are included in the approximation space. Extensions to other systems of linear second-order partial differential equations are discussed. Numerical simulations show only moderate deterioration of the effectivity index for a Poisson ratio close to .

Abstract. We prove convergence and quasi-optimality of a lowest-order adaptive boundary element method for a weakly-singular integral equation in 2D. The adaptive mesh-refinement is driven by the weighted-residual error estimator. By proving that this estimator is not only reliable, but under some regularity assumptions on the given data also efficient on locally refined meshes, we characterize the approximation class in terms of the Galerkin error only. In particular, this yields that no adaptive strategy can do better, and the weighted-residual error estimator is thus an optimal choice to steer the adaptive mesh-refinement. As a side result, we prove a weak form of the saturation assumption.

Abstract. We develop a new analysis for residual-type a posteriori error estimation for a class of highly indefinite elliptic boundary value problems by considering the Helmholtz equation at high wavenumber as our model problem. We employ a classical conforming Galerkin discretization by using hp -finite elements. In [Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp., 79 (2010), pp. 1871–1914], Melenk and Sauter introduced an hp -finite element discretization which leads to a stable and pollution-free discretization of the Helmholtz equation under a mild resolution condition which requires only degrees of freedom, where denotes the spatial dimension. In the present paper, we will introduce an a posteriori error estimator for this problem and prove its reliability and efficiency. The constants in these estimates become independent of the, possibly, high wavenumber provided the aforementioned resolution condition for stability is satisfied. We emphasize that, by using the classical theory, the constants in the a posteriori estimates would be amplified by a factor k .

Abstract. A multilevel augmentation method is considered to solve parameter identification problems in elliptic systems. With the help of the natural linearization technique, the identification problems can be transformed into a linear ill-posed operation equation, where noise exists not only in RHS data but also in operators. Based on multiscale decomposition in solution space, the multilevel augmentation method leads to a fast algorithm for solving discretized ill-posed problems. Combining with Tikhonov regularization, in the implementation of the multilevel augmentation method, one only needs to invert the same matrix with a relatively small size and perform a matrix-vector multiplication at the linear computational complexity. As a result, the computation cost is dramatically reduced. The a posteriori regularization parameter choice rule and the convergence rate for the regularized solution are also studied in this work. Numerical tests illustrate the proposed algorithm and the theoretical estimates.