Publications

2002

Frauenfelder, P. and Lage, C. Concepts - An object-oriented software package for partial differential equations. ESAIM: !M2AN, 36(5), 937-951, 2002. DOI Abstract: Object oriented design has proven itself as a powerful tool in the field of scientific computing. Several software packages, libraries and toolkits exist, in particular in the FEM arena that follow this design methodology providing extensible, reusable, and flexible software while staying competitive to traditionally designed point tools in terms of efficiency. However, the common approach to identify classes is to turn data structures and algorithms of traditional implementations into classes such that the level of abstraction is essentially not raised. In this paper we discuss an alternative way to approach the design challenge which we call “concept oriented design”. We apply this design methodology to Petrov-Galerkin methods leading to a class library for both, boundary element methods (BEM) and finite element methods (FEM). We show as a particular example the implementation of hp-FEM using the library with special attention to the handling of inconsistent meshes.

1998

Lage, C. Concept oriented design of numerical software. SAM Report 1998-07, Seminar for Applied Mathematics, ETH Zürich, 1998. PDF Abstract: The continuously growing computing power of modern computers admits to tackle numerical problems of extreme complexity. This complexity carries over to the numerical methods applied to solve the problems. Whereas the mathematical formulation of these methods does not raise any difficulties, their implementation turns out to be the bottleneck in the realization of numerical applications. In the last years, in order to afford relief, object oriented methods were applied to promote reusable and extensible numerical software, since this kind of flexibility is the key to manage complexity. It became evident that a carefully chosen modularization of the considered methods is a necessary requirement to provide flexible software components. In this paper we give a brief review of object oriented methods to identify the key issues that support a flexible software design and discuss a modularization technique based on mathematical concepts. Finally, the application of this concept oriented approach to boundary element methods is presented.

2014

Engström, C. Spectral approximation of quadratic operator polynomials arising in photonic band structure calculations. Num. Math., 126(3): 413-440, 2014.DOI Galerkin spectral approximation theory for non-self-adjoint quadratic operator polynomials with periodic coefficients is considered. The main applications are complex band structure calculations in metallic photonic crystals, periodic waveguides, and metamaterials. We show that the spectrum of the considered operator polynomials consists of isolated eigenvalues of finite multiplicity with a nonzero imaginary part. The spectral problem is equivalent to a non-compact block operator matrix and norm convergence is shown for a block operator matrix having the same generalized eigenvectors as the original operator. Convergence rates of finite element discretizations are considered and numerical experiments with the p-version and the h-version of the finite element method confirm the theoretical convergence rates.

Klindworth, D. and Schmidt, K. An efficient calculation of photonic crystal band structures using Taylor expansions. Accepted for publication in Commun. Comput. Phys., available as Matheon Preprint #1068, 2014. PDF In this paper we used the same matrices as produced for the paper with Sonia Fliss on Dirichlet-to-Neumann transparent boundary conditions for photonic crystal wave-guides (see below). We exported these matrices to Matlab's binary format using Concepts's concepts::MatfileIO class. All post-processing is then done with Matlab.

Schmidt, K. and Thöns-Zueva, A. Impedance boundary conditions for acoustic time harmonic wave propagation in viscous gases. Preprint series of the Institute of Mathematics 6-2014, Technische Universität Berlin, 2014. PDF

Schmidt, K. and Hiptmair, R. Asymptotic boundary element methods for thin conducting sheets in two dimensions. IEEE Trans. Magn., 50: 469-472, 2014. DOI, PDF

Schmidt, K. and Chernov, A. Robust transmission conditions of high order for thin conducting sheets in two dimensions. IEEE Trans. Magn., 50(2): 41-44, 2014. DOI, PDF

Klindworth, D. and Schmidt, K. Dirichlet-to-Neumann transparent boundary conditions for photonic crystal wave-guides. IEEE Trans. Magn., 50: 217-220, 2014. DOI, PDF

Schmidt, K. and Heier, C. An analysis of Feng's and other symmetric local absorbing boundary conditions. Accepted for publication in ESAIM: !M2AN, 2014. In this article we use a continuous FEM discretisation for local absorbing boundary conditions which involve derivatives of order 4 and higher. For this we use an interior penalty discretisation which additional terms on the nodes on FE mesh on the boundary.

Schmidt, K., Thöns-Zueva, A. and Joly, P. Asymptotic analysis for acoustics in viscous gases close to rigid walls. Math. Models Meth. Appl. Sci., 24(9): 1823-1855, 2014. DOI

Fliss, S., Klindworth, D. and Schmidt, K. Robin-to-Robin transparent boundary conditions for the computation of guided modes in photonic crystal wave-guides. Submitted to BIT, 2014. As for the project with Dirichlet-to-Neumann operators (see below) we used Concepts to construct finite element spaces on coarse meshes that perfectly resolve the circular holes in the photonic crystal unit cells. In addition to the project with Dirichlet-to-Neumann we have to implement mixed variational formulations for which we need finite element subspaces the trace spaces and, if the material cofficient jumps at the boundaries, we additionally need their corresponding dual spaces to cope for the lower regularity of the Neumann and Robin traces at the boundaries. To this end, we employed Concepts' hp2D::TraceSpace and hp1D::DualSpace classes.

Klindworth, D., Schmidt, K. and Fliss, S. Numerical realization of Dirichlet-to-Neumann transparent boundary conditions for photonic crystal wave-guides. Comput. Math. Appl., 67(4): 918-943, 2014. DOI, PDF For the exact computation of guided modes in photonic crystal wave-guides we employ Dirichlet-to-Neumann operators. These operators are computed with the help of Dirichlet problems in a unit cell of the photonic crystal. We used Concepts to construct finite element spaces on coarse meshes with curved cells and large polynomial degrees (greater or equal to five). Concepts is ideal for this purpose as it offers the construction of periodic spaces and it allows for curved cell boundaries, which is important to perfectly resolve the holes/rods (grey circles in the image on the right hand side). Moreover, it is straightforward with Concepts to construct the space in a way that the degrees of freedom are sorted for your needs. In this context it is for example beneficial if one can easily access the degrees of freedom on the boundaries of the unit cell. To this end, the space is constructed such that the first degrees of freedom lie on the boundaries.

Engström, C. Spectral approximation of quadratic operator polynomials arising in photonic band structure calculations. Num. Math., 126(3): 413-440, 2014. DOI For the band structure calculation for frequency dependent material the quadratic eigenvalue problem in the wave vector k as well as its finite element approximation is analysed. Numerical experiments has been performed with Concepts in 2D using p-FEM with curved quadrilateral cells, where exponential convergence of the wave vectors has been achieved.

2013

Schmidt, K. and Hiptmair, R. Asymptotic boundary element methods for thin conducting sheets. Preprint series of the Institute of Mathematics 15-2013, Technische Universität Berlin, 2013. PDF

Wang, M., Schmidt, K., Alparslan, A. and Hafner, C. hp-FEM and PML analysis of plasmonic particles in layered media. Prog. Electromagn. Res., 142: 523-544, 2013. PDF

Schmidt, K. and Chernov, A. A unified analysis of transmission conditions for thin conducting sheets in the time-harmonic eddy current model. SIAM J. Appl. Math, 73(6): 1980-2003, 2013. DOI, PDF

Claeys X., Hiptmair, R., and Spindler, E. A second-kind Galerkin boundary element method for scattering at composite objects. SAM Report 2013-13 (revised), Seminar for Applied Mathematics, ETH Zürich, 2013. PDF Concepts was used to obtain reliable reference solutions by <em>hp</em>-adaptive FEM discretisation for the scattering problem on curved multi-material bodies with corner singularities. The solution computed on the adaptive meshes is evaluated on a uniformly distributed set of points on the material interfaces.

2011

Schmidt, K. and Tordeux, S. High order transmission conditions for thin conductive sheets in magneto-quasistatics. ESAIM: !M2AN, 45(6): 1115-1140, 2011. PDF

Wang, M., Engström, C., Schmidt, K. and Hafner, C. On high-order FEM applied to canonical scattering problems in plasmonics. J. Comput. Theor. Nanosci., 8(8): 1-9, 2011. PDF

Brandsmeier, H., Schmidt, K. and Schwab, C. A multiscale hp-FEM for 2D photonic crystal bands. J. Comput. Phys., 230(2): 349-374, 2011. DOI, PDF

2010

Schmidt, K. and Kappeler, R. Efficient computation of photonic crystal waveguide modes with dispersive material. Optics Express, 18(7): 7307-7322, 2010. PDF

Schmidt, K. and Tordeux, S. Asymptotic modelling of conductive thin sheets. Z. Angew. Math. Phys., 61(4): 603-626, 2010. DOI, PDF

2009

Schmidt, K. and Kauf, P. Computation of the band structure of two-dimensional photonic crystals with hp finite elements. Comp. Meth. App. Mech. Engr., 198: 1249-1259, 2009. PDF

Engström, C., Hafner, C. and Schmidt, K. Computations of lossy Bloch waves in two-dimensional photonic crystals. J. Comput. Theor. Nanosci., 6: 775-783, 2009. PDF

2008

Schmidt, K., Sterz, O. and Hiptmair, R. Estimating the eddy-current modelling error. IEEE Trans. Magn., 44: 686-689, 2008. PDF