This project aims to investigate the discretization of elliptic partial differential equations with periodic boundary conditions on anisotropic spaces of translates. Of special interest are the influence of the anisotropic sampling lattice, the choice of the translation invariant space, convergence results and the investigation of periodic wavelets.
References

Bergmann, R and Merkert, D (2017). FFTbased homogenization on periodic anisotropic translation invariant spaces
In this paper we derive a discretisation of the equation of quasistatic elasticity in homogenization in form of a variational formulation and the socalled LippmannSchwinger equation, in anisotropic spaces of translates of periodic functions. We unify and extend the truncated Fourier series approach, the constant finite element ansatz and the anisotropic lattice derivation. The resulting formulation of the LippmannSchwinger equation in anisotropic translation invariant spaces unifies and analyses for the first time both the Fourier methods and finite element approaches in a common mathematical framework. We further define and characterize the resulting periodised Green operator. This operator coincides in case of a Dirichlet kernel corresponding to a diagonal matrix with the operator derived for the Galerkin projection stemming from the truncated Fourier series approach and to the anisotropic lattice derivation for all other Dirichlet kernels. Additionally, we proof the boundedness of the periodised Green operator. The operator further constitutes a projection if and only if the space of translates is generated by a Dirichlet kernel. Numerical examples for both the de la Vallée Poussin means and Box splines illustrate the flexibility of this framework.@online{BM17, author = {Bergmann, Ronny and Merkert, Dennis}, title = {{FFT}based homogenization on periodic anisotropic translation invariant spaces}, year = {2017}, eprint = {1701.04685}, eprinttype = {arXiv} }

Bergmann, R and Merkert, D (2017). Approximation of periodic PDE solutions with anisotropic translation invariant spaces. 2017 International Conference on Sampling Theory and Applications (SampTA). 396–9
We approximate the quasistatic equation of linear elasticity in translation invariant spaces on the torus. This unifies different FFTbased discretisation methods into a common framework and extends them to anisotropic lattices. We analyse the connection between the discrete solution spaces and demonstrate the numerical benefits. Finite element methods arise as a special case of periodised Box spline translates.@inproceedings{BM17a, author = {Bergmann, Ronny and Merkert, Dennis}, title = {Approximation of periodic PDE solutions with anisotropic translation invariant spaces}, year = {2017}, pages = {396399}, booktitle = {2017 International Conference on Sampling Theory and Applications (SampTA)}, doi = {10.1109/SAMPTA.2017.8024347} }