We present, on the simplest possible case, what we consider as the very basic features of the (brand new) virtual element method. As the readers will easily recognize, the virtual element method could easily be regarded as the ultimate evolution of the mimetic finite differences approach. However, in their last step they became so close to the traditional finite elements that we decided to use a different perspective and a different name. Now the virtual element spaces are just like the usual finite element spaces with the addition of suitable non-polynomial functions. This is far from being a new idea. See for instance the very early approach of E. Wachspress [A Rational Finite Element Basic (Academic Press, 1975)] or the more recent overview of T.-P. Fries and T. Belytschko [The extended/generalized finite element method: An overview of the method and its applications, Int. J. Numer. Methods Engrg.84 (2010) 253–304]. The novelty here is to take the spaces and the degrees of freedom in such a way that th...

Basic principles of Virtual Element Methods

This article considers the problem of approximating a general asymptotically smooth function in two variables, typically arising in integral formulations of boundary value problems, by a sum of products of two functions in one variable. From these results an iterative algorithm for the low-rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed. This algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance.

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Approximation of boundary element matrices

Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory

Molecular Dynamics.- Event-Driven Molecular Dynamics.- Direct Simulation Monte Carlo.- Rigid-Body Dynamics.- Cellular Automata.- Bottom-to-Top Reconstruction.- Brownian Dynamics for the Simulation of Granular Flows.

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Computational Granular Dynamics: Models and Algorithms

This paper presents the adaptive cross approximation (ACA) algorithm to reduce memory and CPU time overhead in the method of moments (MoM) solution of surface integral equations. The present algorithm is purely algebraic; hence, its formulation and implementation are integral equation kernel (Green's function) independent. The algorithm starts with a multilevel partitioning of the computational domain. The interactions of well-separated partitioning clusters are accounted through a rank-revealing LU decomposition. The acceleration and memory savings of ACA come from the partial assembly of the rank-deficient interaction submatrices. It has been demonstrated that the ACA algorithm results in O(NlogN) complexity (where N is the number of unknowns) when applied to static and electrically small electromagnetic problems. In this paper the ACA algorithm is extended to electromagnetic compatibility-related problems of moderate electrical size. Specifically, the ACA algorithm is used to study compact-range ground planes and electromagnetic interference and shielding in vehicles. Through numerical experiments, it is concluded that for moderate electrical size problems the memory and CPU time requirements for the ACA algorithm scale as N/sup 4/3/logN.

The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems

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.

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Adaptive low-rank approximation of collocation matrices

Boundary Integral Equations.- Boundary Element Methods.- Approximation of Boundary Element Matrices.- Implementation and Numerical Examples.

The Fast Solution of Boundary Integral Equations

It is well known that the classical boundary-element method (BEM) yields fully populated matrices. Their manipulation is cumbersome with respect to memory consumption and computational costs. This paper describes a novel approach where the matrices are split into collections of blocks of various sizes. Those blocks which describe remote interactions are adaptively approximated by low rank submatrices. This procedure reduces the algorithmic complexity for matrix setup and matrix-by-vector products to approximately O(N). The proposed method has been examined in a testing environment and implemented into an existing BEM-finite-element method (FEM) code for electromagnetic and electromechanical problems. The advantages of the new method are demonstrated by means of several examples.

The adaptive cross-approximation technique for the 3D boundary-element method

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 $\ensuremath{\psi}$, which acts as the potential for the rotation free part of the dielectric displacement field $\mathbf{D}$. 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 $\ensuremath{\psi}$ and the electrostatic potential $\ensuremath{\phi}$ reduce to a system of coupled partial differential equations. The approach is illustrated by its application to simple geometries.

Novel formulation of nonlocal electrostatics.

A special form of the Boltzmann collision operator for the hard spheres model is introduced. The possibilities of fast numerical computation of the collision operator based on this form and the Fast Fourier Transform are discussed. A new difference scheme for the Boltzmann equation for the hard spheres model is developed. The results of some numerical tests and accuracy comparisons with the Direct Simulation Monte Carlo (DSMC) method are presented.

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Sergej Rjasanow

Papers

Adaptive low-rank approximation of collocation matrices

The Fast Solution of Boundary Integral Equations

The adaptive cross-approximation technique for the 3D boundary-element method

Novel formulation of nonlocal electrostatics.

Fast deterministic method of solving the Boltzmann equation for hard spheres