We present an efficient method to compute the electrostatic fields, torques and forces in dipolar systems, which is based on the fast Fourier transform for nonequispaced data (NFFT). We consider 3d-periodic, 2d-periodic, 1d-periodic as well as 0d-periodic (open) boundary conditions. The method is based on the corresponding Ewald formulas, which immediately lead to an efficient algorithm only in the 3d-periodic case. In the other cases we apply the NFFT based fast summation in order to approximate the contributions of the nonperiodic dimensions in Fourier space. This is done by regularizing or periodizing the involved functions, which depend on the distances of the particles regarding the nonperiodic dimensions. The final algorithm enables a unified treatment of all types of periodic boundary conditions, for which only the precomputation step has to be adjusted.

We present an efficient method to compute the electrostatic fields, torques and forces in dipolar systems, which is based on the fast Fourier transform for nonequispaced data (NFFT). We consider 3d-periodic, 2d-periodic, 1d-periodic as well as 0d-periodic (open) boundary conditions. The method is based on the corresponding Ewald formulas, which immediately lead to an efficient algorithm only in the 3d-periodic case. In the other cases we apply the NFFT based fast summation in order to approximate the contributions of the nonperiodic dimensions in Fourier space. This is done by regularizing or periodizing the involved functions, which depend on the distances of the particles regarding the nonperiodic dimensions. The final algorithm enables a unified treatment of all types of periodic boundary conditions, for which only the precomputation step has to be adjusted.

The present thesis is dedicated to the efficient computation of electrostatic interactions in particle systems, which is of great importance in the field of molecular dynamics simulations. In order to compute the therefor required physical quantities with only O(N log N) arithmetic operations, so called particle-mesh methods make use of the well-known Ewald summation approach and the fast Fourier transform (FFT). Typically, such methods are able to handle systems of point charges subject to periodic boundary conditions in all spatial directions. However, periodicity is not always desired in all three dimensions and, moreover, also interactions to dipoles play an important role in many applications. Within the scope of the present work, we consider the particle-particle NFFT method (P²NFFT), a particle-mesh approach based on the fast Fourier transform for nonequispaced data (NFFT). An extension of this method for mixed periodic as well as open boundary conditions is presented. Furthermore, the method is appropriately modified in order to treat particle systems containing both charges and dipoles. Consequently, an efficient algorithm for mixed charge-dipole systems, that additionally allows a unified handling of various types of periodic boundary conditions, is presented for the first time. Appropriate error estimates as well as parameter tuning strategies are developed and verified by numerical examples. / Die vorliegende Arbeit widmet sich der Berechnung elektrostatischer Wechselwirkungen in Partikelsystemen, was beispielsweise im Bereich der molekulardynamischen Simulationen eine zentrale Rolle spielt. Um die dafür benötigten physikalischen Größen mit lediglich O(N log N) arithmetischen Operationen zu berechnen, nutzen sogenannte Teilchen-Gitter-Methoden die Ewald-Summation sowie die schnelle Fourier-Transformation (FFT). Typischerweise können derartige Verfahren Systeme von Punktladungen unter periodischen Randbedingungen in allen Raumrichtungen handhaben. Periodizität ist jedoch nicht immer bezüglich aller drei Dimensionen erwünscht. Des Weiteren spielen auch Wechselwirkungen zu Dipolen in vielen Anwendungen eine wichtige Rolle. Zentraler Gegenstand dieser Arbeit ist die Partikel-Partikel-NFFT Methode (P²NFFT), ein Teilchen-Gitter-Verfahren, welches auf der schnellen Fouriertransformation für nichtäquidistante Daten (NFFT) basiert. Eine Erweiterung dieses Verfahrens auf gemischt periodische sowie offene Randbedingungen wird vorgestellt. Außerdem wird die Methode für die Behandlung von Partikelsystemen, in denen sowohl Ladungen als auch Dipole vorliegen, angepasst. Somit wird erstmalig ein effizienter Algorithmus für gemischte Ladungs-Dipol-Systeme präsentiert, der zusätzlich die Behandlung sämtlicher Arten von Randbedingungen mit einem einheitlichen Zugang erlaubt. Entsprechende Fehlerabschätzungen sowie Strategien für die Parameterwahl werden entwickelt und anhand numerischer Beispiele verifiziert.