Christoph Pflaum

Error analysis of the combination technique

Summary. The combination technique is a method to reduce the computational time in the numerical approximation of partial differential equations. In this paper, we present a new technique to analyze the convergence rate of the combination technique. This technique is applied to general second order elliptic differential equations in two dimensions. Furthermore, it is proved that the combination technique for Poissons equation convergences in arbitrary dimensions.

Numerical simulation of light propagation in silver nanowire films using time-harmonic inverse iterative method

The interaction between light and silver nanowires (Ag NWs) in a thin film is simulated by solving Maxwells equations numerically. Time-harmonic inverse iterative method is implemented to overcome the problem of negative permittivity of silver, which makes the classical finite-difference time-domain iteration unstable. The method is validated by showing the correspondence between the plasmonic resonance of an Ag NW from a two dimensional simulation and the analytical solution. In agreement with previous experimental studies, the simulation results show that the transmissivity of the Ag NW films is higher than expected from the geometric aperture. The cause of this phenomenon is studied using TE/TM modes analysis for Ag NW films with different surface coverage of parallel-aligned Ag NWs. Furthermore, 3D simulation of Ag NW films with randomly arranged Ag NWs is performed by parallel computation on high performance computers. A binder layer is taken into account for a preliminary comparison between the sim...

Convergence of the Combination Technique for Second-Order Elliptic Differential Equations

The combination technique is an algorithm for the approximate solution of partial differential equations on sparse grids that has to be combined with a suitable standard discretization. The advantage of the combination technique compared to the standard discretization is that the same accuracy is achieved with many fewer grid points. In this paper, the combination technique is used with a bilinear finite element discretization. Depending on the smoothness of the solution and the coefficients, it is proved for general second-order elliptic differential equations on the unit square that the combined solution converges with order O(h) or O(h log h-1) in the energy norm and with order O(h2 log h-1) or O(h3/2) in the L2-norm, respectively. This holds even if the bilinear form corresponding to the elliptic equation is not symmetric positive definite. The proof does not use an asymptotic error expansion, but Sobolev space techniques.

Semi-unstructured grids

Abstract We present a novel automatic grid generator for the finite element discretization of partial differential equations in 3D. The grids constructed by this grid generator are composed of a pure tensor product grid in the interior of the domain and an unstructured grid which is only contained in boundary cells. The unstructured component consists of tetrahedra, each of which satisfies a maximal interior angle condition. By suitable constructing the boundary cells, the number of types of boundary subcells is reduced to 12 types. Since this grid generator constructs large structured grids in the interior and small unstructured grids near the boundary, the resulting semi-unstructured grids have similar properties as structured tensor product grids. Some appealing properties of this method are computational efficiency and natural construction of coarse grids for multilevel algorithms. Numerical results and an analysis of the discretization error are presented.

Third-dimensional finite element computation of laser cavity eigenmodes

A new method for computing eigenmodes of a laser resonator by the use of finite element analysis is presented. For this purpose, the scalar wave equation (Δ + k2)Ẽ(x, y, z) = 0 is transformed into a solvable three-dimensional eigenvalue problem by the separation of the propagation factor exp(-ikz) from the phasor amplitude Ẽ(x, y, z) of the time-harmonic electrical field. For standing wave resonators, the beam inside the cavity is represented by a two-wave ansatz. For cavities with parabolic optical elements, the new approach has successfully been verified by the use of the Gaussian mode algorithm. For a diode-pumped solid-state laser with a thermally lensing crystal inside the cavity, the expected deviation between Gaussian approximation and numerical solution could be demonstrated clearly.

Fast Expression Templates

Expression templates (ET) can significantly reduce the implementation effort of mathematical software. For some compilers, especially for those of supercomputers, however, it can be observed that classical ET implementations do not deliver the expected performance. This is because aliasing of pointers in combination with the complicated ET constructs becomes much more difficult. Therefore, we introduced the concept of enumerated variables, which are provided with an additional integer template parameter. Based on this new implementation of ET we obtain a C++ code whose performance is very close to the handcrafted C code. The performance results of these so-called Fast ET are presented for the Hitachi SR8000 supercomputer and the NEC SX6, both with automatic vectorization and parallelization. Additionally we studied the combination of Fast ET and OpenMP on a high performance Opteron cluster.

Robust Convergence of Multilevel Algorithms for Convection-Diffusion Equations

A new approach is developed to analyze convergence of multilevel algorithms for convection-diffusion equations. This approach uses a multilevel recursion formula, which can be applied to a variety of nonsymmetric problems. Here, the recursion formula is applied to a robust multilevel algorithm for convection-diffusion equations with convection in the x- or y-direction. The multilevel algorithm uses semicoarsening, line relaxation, and prewavelets. The convergence rate is proved to be less than 0.18 independent of the size of the convection term and the number of unknowns. The assumptions allow the convection term to have a turning point, so that an interior layer can appear in the solution of the convection-diffusion equation. The computational cost of the multilevel cycle is about O(N log N) independent of the size of the convection term, where N is the number of unknowns. It is proved that O(log N) multilevel cycles starting from the initial guess 0 lead to an O(N-2) algebraic error with respect to the

Discretization of elliptic differential equations on curvilinear bounded domains with sparse grids

L^\infty

New Finite Elements for Large-Scale Simulation of Optical Waves

norm, independent of the size of the convection term.

Advanced expression templates programming

Elliptic differential equations can be discretized with bilinear finite elements. Using sparse grids instead of full grids, the dimension of the finite element space for the 2D problem reduces fromO(N2) toO (N logN) while the approximation properties are nearly the same for smooth functions. A method is presented which discretizes elliptic differential equations on curvilinear bounded domains with adaptive sparse grids. The grid is generated by a transformation of the domain. This method has the same behaviour of convergence like the sparse grid discretization on the unit square.ZusammenfassungElliptische Differentialgleichungen werden häufig mittels bilinearer Finite-Elemente diskretisiert. Verwendet man hierfür dünne Gitter anstelle voller Gitter, so reduziert sich die Dimension des Finite-Element-Raumes eines 2D-Problems vonO(N2) aufO(N logN). Die Approximationseigenschaften bleiben jedoch nahezu erhalten. Es wird eine Methode zur Diskretisierung elliptischer Differentialgleichungen auf krummlinig berandeten Gebieten mittels adaptiver dünner Gitter vorgestellt. Das Gitter wird durch eine Transformation des Gebietes erzeugt. Das Konvergenzverhalten dieses Verfahrens kommt dem der Dünngitter-Diskretisierung auf dem Einheitsquadrat gleich.