We consider the Maxwell equations in a domain with Lipschitz boundary and the boundary integral operator A occuring in the Calderón projector. We prove an inf-sup condition for A using a Hodge decomposition. We apply this to two types of boundary value problems: the exterior scattering problem by a perfectly conducting body, and the dielectric problem with two different materials in the interior and exterior domain. In both cases we obtain an equivalent boundary equation which has a unique solution. We then consider Galerkin discretizations with Raviart-Thomas spaces. We show that these spaces have discrete Hodge decompositions which are in some sense close to the continuous Hodge decomposition. This property allows us to prove quasioptimal convergence of the resulting boundary element methods.

Abstract. Arbitrage-free prices u of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equationThis PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for A can be replaced by a sparse matrix in the wavelet basis, and the linear systems in each implicit time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for M time steps is bounded by O (MN(log(N )) 2 ) operations and O(N log(N )) memory. The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solution in the same complexity as finite difference approximations of the standard Black-Scholes equation. Computational examples for various Lévy price processes are presented.Mathematics Subject Classification. 65N30, 60J75.

This paper presents a theoretical discussion as well as novel solution algorithms for problems of scattering on smooth two-dimensional domains under Zaremba boundary conditions-for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the proposed numerical methods, which is provided for the first time in the present contribution, concerns detailed information about the singularity structure of solutions of the Helmholtz operator under boundary conditions of Zaremba type. The new numerical method is based on use of Green functions and integral equations, and it relies on the Fourier Continuation method for regularization of all smooth-domain Zaremba singularities as well as newly derived quadrature rules which give rise to high-order convergence even around Zaremba singular points. As demonstrated in this paper, the resulting algorithms enjoy high-order convergence, and they can be used to efficiently solve challenging Helmholtz boundary value problems and Laplace eigenvalue problems with high-order accuracy.

Abstract.We consider the numerical solution of diffusion problems in (0, T ) × Ω for Ω ⊂ R d and for T > 0 in dimension d ≥ 1. We use a wavelet based sparse grid space discretization with meshwidth h and order p ≥ 1, and hp discontinuous Galerkin time-discretization of order r = O(|log h|) on a geometric sequence of O(|log h|) many time steps. The linear systems in each time step are solved iteratively by O(|log h|) GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives anwhere N is the total number of operations, provided that the initial data satisfies u0 ∈ H ε (Ω) with ε > 0 and that u(x, t) is smooth in x for t > 0. Numerical experiments in dimension d up to 25 confirm the theory.Mathematics Subject Classification. 65N30.

Communicated by E. MeisterThe solution of the three-dimensional mixed boundary value problem for the Laplacian in a polyhedral domain has special singular forms at corners and edges. A 'tensor-product' decomposition of those singular forms along the edges is derived. We present a strongly elliptic system of boundary integral equations which is equivalent to the mixed boundary value problem. Regularity results for the solution of this system of integral equations are given which allow us to analyse the influence of graded meshes on the rate of convergence or the corresponding boundary element Galerkin solutions. We show that it suffices to refine the mesh only towards the edges of the surfaces to regain the optimal rate of convergence. IntroductionWe consider the mixed Dirichlet-Neumann problem (1.1) for harmonic functions in a polyhedral domain Q whose boundary r = r1 u r2 = us= rj consists of plane faces I-". We derive regularity results for the solution when the inhomogeneous boundary data u = g , on rl and au/an=g2 on Ft satisfy g l = G l r , and gt= (aG/an)lr,, where G belongs to the Sobolev space H'+"(Q), s20, with AGEH~((R), q = max (0, s -1). In this case the original problem (1.1) can be converted into a mixed boundary value problem (2.3) for the Poisson equation with vanishing boundary data. Our main results are decompositions for the solution of (1.1) into sums of edge and corner singularities and regular parts. We prove those regularity results by first analysing the solution of problem (2.3) with the method of Dauge [3], [4]. Then we apply our refined analysis from [14] and we obtain for the solution of (2.3) a decomposition where all appearing singularity functions are of tensor-product form. Returning to problem (1.1) we then arrive at the desired results for its solution. Taking traces on the boundary r we therefore obtain corresponding decompositions for the unknown Cauchy data of an equivalent system of boundary integral equations (1.7).