This article deals with the solution of integral equations using collocation methods with almost linear complexity. Methods such as fast multipole, panel clustering and H-matrix methods gain their efficiency from approximating the kernel function. The proposed algorithm which uses the H-matrix format, in contrast, is purely algebraic and relies on a small part of the collocation matrix for its blockwise approximation by low-rank matrices. Furthermore, a new algorithm for matrix partitioning that significantly reduces the number of blocks generated is presented.

The accurate modeling of the dielectric properties of water is crucial for many applications in physics, computational chemistry, and molecular biology. This becomes possible in the framework of nonlocal electrostatics, for which we propose a novel formulation allowing for numerical solutions for the nontrivial molecular geometries arising in the applications mentioned before. Our approach is based on the introduction of a secondary field psi, which acts as the potential for the rotation free part of the dielectric displacement field D. For many relevant models, the dielectric function of the medium can be expressed as the Green's function of a local differential operator. In this case, the resulting coupled Poisson (-Boltzmann) equations for psi and the electrostatic potential phi reduce to a system of coupled partial differential equations. The approach is illustrated by its application to simple geometries.

No abstract

No abstract

The BEM-based finite element method is reviewed and extended with higher order basis functions on general polygonal meshes. These functions are defined implicitly as local solutions of the underlying homogeneous problem with constant coefficients. They are treated by means of boundary integral formulations and are approximated using the boundary element method in the numerics. To obtain higher order convergence, a new approximation of the material coefficient is proposed since previous strategies are not sufficient. Following recent ideas, error estimates are proved which guarantee quadratic convergence in the H 1 -norm and cubic convergence in the L 2 -norm. The numerical realization is discussed and several experiments confirm the theoretical results. Introduction.In the field of numerical methods for partial differential equations there is an increasing interest in nonsimplicial meshes. Several applications in solid mechanics, biomechanics, and geological science show a need for general elements within a finite element simulation. Discontinuous Galerkin methods [9] and mimetic finite difference methods [3] are able to handle such meshes. Nevertheless, these two strategies yield nonconforming approximations in their original versions which are not in the Sobolev space in which the exact solution lies.In 1975, Wachspress [26] proposed the construction of conforming rational basis functions on convex polygons with any number of sides. In recent years, several improved basis functions on polygonal elements have been introduced and applied in linear elasticity [24] or computer graphics [15,17], for example. There are even the first attempts which seek to introduce quadratic finite elements on polygons [20]. The recent publication [2] extends the nodal mimetic discretization strategy to arbitrary order on polygonal meshes.In [7] and the detailed work [6], a new kind of conforming finite element method on polygonal meshes has been proposed which uses basis functions that fulfill the differential equation locally. In the local problems constant material parameters and vanishing right-hand sides are prescribed. In case of the diffusion equation harmonic basis functions are recovered. This method has been studied in several articles concerning convergence [12,13] and residual error estimates for adaptive mesh refinement [27]. In the following, the theory is extended to a higher order method on polygonal meshes.The outline of this article is as follows. In section 2, some notation as well as a model problem are introduced. We review the first order method proposed in [7] and give a definition for regular meshes. The BEM-based FEM is extended to a higher order scheme in section 3. Afterward, we introduce interpolation operators in section 4 and prove interpolation properties which yield error estimates for the finite element

Electrostatic interactions play a crucial role in many biomolecular processes, including molecular recognition and binding. Biomolecular electrostatics is modulated to a large extent by the water surrounding the molecules. Here, we present a novel approach to the computation of electrostatic potentials which allows the inclusion of water structure into the classical theory of continuum electrostatics. Based on our recent purely differential formulation of nonlocal electrostatics [Hildebrandt, et al. (2004) Phys. Rev. Lett., 93, 108104] we have developed a new algorithm for its efficient numerical solution. The key component of this algorithm is a boundary element solver, having the same computational complexity as established boundary element methods for local continuum electrostatics. This allows, for the first time, the computation of electrostatic potentials and interactions of large biomolecular systems immersed in water including effects of the solvent's structure in a continuum description. We illustrate the applicability of our approach with two examples, the enzymes trypsin and acetylcholinesterase. The approach is applicable to all problems requiring precise prediction of electrostatic interactions in water, such as protein-ligand and protein-protein docking, folding and chromatin regulation. Initial results indicate that this approach may shed new light on biomolecular electrostatics and on aspects of molecular recognition that classical local electrostatics cannot reveal.

No abstract

In this work, we propose a new Galerkin-Petrov method for the numerical solution of the classical spatially homogeneous Boltzmann equation. This method is based on an approximation of the distribution function by associated Laguerre polynomials and spherical harmonics and test an a variational manner with globally defined threedimensional polynomials. A numerical realization of the algorithm is presented. The algorithmic developments are illustrated with the help of several numerical tests.