NAFeb 26, 2019
Error estimates in balanced norms of finite element methods for higher order reaction-diffusion problemsSebastian Franz, Hans-G. Roos
Error estimates of finite element methods for reaction-diffusion problems are often realised in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the $H^m$ seminorm for $2m$-th order problems leads to a balanced norm which reflects the layer behaviour correctly. We prove error estimates in such balanced norms and improve thereby existing estimates known in literature.
NAMay 31, 2011
On the sharpness of Green's function estimates for a convection-diffusion problemSebastian Franz, Natalia Kopteva
Linear singularly perturbed convection-diffusion problems with characteristic layers are considered in three dimensions. We demonstrate the sharpness of our recently obtained upper bounds for the associated Green's function and its derivatives in the $L_1$ norm. For this, in this paper we establish the corresponding lower bounds. Both upper and lower bounds explicitly show any dependence on the singular perturbation parameter.
NANov 8, 2016
Numerical methods for changing type systemsSebastian Franz, Sascha Trostorff, Marcus Waurick
In this note we develop a numerical method for partial differential equations with changing type. Our method is based on a unified solution theory found by Rainer Picard for several linear equations from mathematical physics. Parallel to the solution theory already developed, we frame our numerical method in a discontinuous Galerkin approach in space-time with certain exponentially weighted spaces.
APNov 23, 2017
Resolvent estimates and numerical implementation for the homogenisation of one-dimensional periodic mixed type problemsSebastian Franz, Marcus Waurick
We study a homogenisation problem for problems of mixed type in the framework of evolutionary equations. The change of type is highly oscillatory. The numerical treatment is done by a discontinuous Galerkin method in time and a continuous Galerkin method in space.
NANov 22, 2016
On a connection between layer-adapted exponentially graded and S-type meshesSebastian Franz, Christos Xenophontos
In this short note we analyse a connection between the exponentially graded and the class of S-type meshes for singularly perturbed problems. As a by-product we obtain a slightly modified and more general class of layer-adapted meshes.
NAOct 2, 2018
Homogenisation of parabolic/hyperbolic mediaSebastian Franz, Marcus Waurick
We consider an evolutionary problem with rapidly oscillating coefficients. This causes the problem to change frequently between a parabolic and an hyperbolic state. We prove convergence of the homogenisation process in the unit square and present a numerical method to deal with approximations of the resulting equations. A numerical study finalises the contribution.