Volume 13, Issue 4

Abstract. We present a detailed survey of discrete functional analysis tools (consistency results, Poincaré and Sobolev embedding inequalities, discrete W 1, p compactness, discrete compactness in space and in time) for the so-called Discrete Duality Finite Volume (DDFV) schemes in three space dimensions. We concentrate mainly on the 3D CeVe-DDFV scheme presented in [IMA J. Numer. Anal., 32 (2012), pp. 1574–1603]. Some of our results are new, such as a general time-compactness result based upon the idea of Kruzhkov (1969); others generalize the ideas known for the 2D DDFV schemes or for traditional two-point-flux finite volume schemes. We illustrate the use of these tools by studying convergence of discretizations of nonlinear elliptic-parabolic problems of Leray–Lions kind, and provide numerical results for this example.

Abstract. We investigate the problem of sparse optimal controls for the so-called Schlögl model and the FitzHugh–Nagumo system. In these reaction–diffusion equations, traveling wave fronts occur that can be controlled in different ways. The L 1 -norm of the distributed control is included in the objective functional so that optimal controls exhibit effects of sparsity. We prove the differentiability of the control-to-state mapping for both dynamical systems, show the well-posedness of the optimal control problems and derive first-order necessary optimality conditions. Based on them, the sparsity of optimal controls is shown. The theory is illustrated by various numerical examples, where wave fronts or spiral waves are controlled in a desired way.

Abstract. Existence and uniqueness of generalized solutions to initial value problems for a class of abstract differential-algebraic equations (DAEs) is shown. The class of equations covers, in particular, the Stokes and Oseen problem describing the motion of an incompressible or nearly incompressible Newtonian fluid but also their spatial semi-discretization. The equations are governed by a block operator matrix with entries that fulfill suitable inf-sup conditions. The problem data are required to satisfy appropriate consistency conditions. The results in infinite dimensions are compared in detail with those known for the DAEs that arise after semi-discretization in space. Explicit solution formulas are derived in both cases.

Abstract. In this paper initial boundary value problems, defined using quasilinear diffusion equations of Volterra type, are considered. These equations arise for instance to describe diffusion processes in viscoelastic media whose behavior is represented by a Voigt–Kelvin model or a Maxwell model. A finite difference discretization defined on a general non-uniform grid with second order convergence order in space is proposed. The analysis does not follow the usual splitting of the global error using the solution of an elliptic equation induced by the integro-differential equation. The new approach enables us to reduce the smoothness required to the theoretical solution when the usual split technique is used. Non-singular and singular kernels are considered. Numerical simulations which show the effectiveness of the method are included.

Abstract. We present some new error estimates for the eigenvalues and eigenfunctions obtained by the Rayleigh–Ritz method, the common variational method to solve eigenproblems. The errors are bounded in terms of the error of the best approximation of the eigenfunction under consideration by functions in the ansatz space. In contrast to the classical theory, the approximation error of eigenfunctions other than the given one does not enter into these estimates. The estimates are based on a bound for the norm of a certain projection operator, e.g., in finite element methods for second order eigenvalue problems, the H 1 -norm of the L 2 -projection onto the finite element space.