Harry Yserentant

Ralf Kornhuber, Daniel Peterseim, Harry Yserentant

Numerical homogenization tries to approximate the solutions of elliptic partial differential equations with strongly oscillating coefficients by functions from modified finite element spaces. We present in this paper a class of such methods that are very closely related to the method of Målqvist and Peterseim [Math. Comp. 83, 2014]. Like the method of Målqvist and Peterseim, these methods do not make explicit or implicit use of a scale separation. Their compared to that in the work of Målqvist and Peterseim strongly simplified analysis is based on a reformulation of their method in terms of variational multiscale methods and on the theory of iterative methods, more precisely, of additive Schwarz or subspace decomposition methods.

Sadegh Jokar, Volker Mehrmann, Marc Pfetsch et al.

A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure is presented that uses linear programming to find a good approximation to the sparse solution on a given refinement level. Then only those parts of the mesh are refined that belong to large expansion coefficients. Error estimates for this procedure are refined and the behavior of the procedure is demonstrated via some simple elliptic model problems.

Stephan Scholz, Harry Yserentant

The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wavefunctions, depend on 3N variables, three spatial dimensions for each electron. We study the approximability of these wavefunctions by linear combinations of anisotropic Gauss functions, or more precisely Gauss-Hermite functions, products of polynomials and anisotropic Gauss functions in the narrow sense. We show that the original, singular wavefunctions can up to given accuracy and a negligibly small residual error be approximated with only insignificantly more such terms than their convolution with a Gaussian kernel of sufficiently small width and that basically arbitrary orders of convergence can be reached. This is a fairly surprising result, since it essentially means that by this type of approximation, the intricate hierarchies of non-smooth cusps in electronic wavefunctions have almost no impact on the convergence, once the global structure is resolved.

Harry Yserentant

Different variants of approximate inverse iteration like the locally optimal block preconditioned conjugate gradient method became in recent years increasingly popular for the solution of the large matrix eigenvalue problems arising from the discretization of selfadjoint elliptic partial differential equations, in particular for the calculation of the minimum eigenvalue. We extend in this little note the classical convergence theory of D'yakonov and Orekhov [Math. Notes 27 (1980)] to the case of operators with an essential spectrum on infinite dimensional Hilbert spaces and allow for arbitrary, sufficiently small perturbations of the solutions of the equation that links the iterates. The note complements the much more elaborate convergence theory of Neymeyr and Knyazev and Neymeyr for the matrix case (see [Knyazev and Neymeyr, SIAM J. Matrix Anal. Appl. 31 (2009)] and the references therein) and is suitable for classroom presentation.