13 citations · 28 across the 16 of their papers we have counts for
18 papers · 1 filter
Post-processing and improved error estimates of numerical methods for evolutionary systems
Sebastian Franz
We consider evolutionary systems, i.e. systems of linear partial differential equations arising from the mathematical physics. For these systems there exists a general solution the…
A convection-diffusion problem with a large shift on Duran meshes
Mirjana Brdar, Sebastian Franz, Hans-Goerg Roos
A convection-diffusion problem with a large shift in space is considered. Numerical analysis of high order finite element methods on layer-adapted Duran type meshes, as well as on…
Pointwise-in-time a posteriori error control for higher-order discretizations of time-fractional parabolic equations
Sebastian Franz, Natalia Kopteva
Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error e…
Numerical analysis of a singularly perturbed convection diffusion problem with shift in space
Mirjana Brdar, Sebastian Franz, Lars Ludwig +1
We consider a singularly perturbed convection-diffusion problem that has in addition a shift term. We show a solution decomposition using asymptotic expansions and a stability resu…
Maximum norm a posteriori error estimates for convection-diffusion problems
Alan Demlow, Sebastian Franz, Natalia Kopteva
We prove residual-type a posteriori error estimates in the maximum norm for a linear scalar elliptic convection-diffusion problem that may be singularly perturbed. Similar error an…
A balanced norm error estimation for the time-dependent reaction-diffusion problem with shift in space
Mirjana Brdar, Sebastian Franz, Lars Ludwig +1
We consider a singularly perturbed time-dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Durán- and S-type we derive a-priori error es…
Singularly perturbed reaction-diffusion problems as first order systems
Sebastian Franz
We consider a singularly perturbed reaction diffusion problem as a first order two-by-two system. Using piecewise discontinuous polynomials for the first component and -co…
Anisotropic -norm error estimates for rectangular -elements
Sebastian Franz
For the discretisation of -functions on rectangular meshes there are at least three families of elements, namely Raviart-Thomas-, Brezzi-Douglas-Marini- and Arnold-Boffi-F…
Mixed FEM for singularly perturbed fourth order boundary value problems with two small parameters
C. Xenophontos, S. Franz, I. Sykopetritou
We consider fourth order singularly perturbed boundary value problems with two small parameters, and the approximation of their solution by the version of the Finite Element M…
Continuous time integration for changing type systems
Sebastian Franz
We consider variational time integration using continuous Galerkin Petrov methods applied to evolutionary systems of changing type. We prove optimal-order convergence of the error…
Error estimates in balanced norms of finite element methods for higher order reaction-diffusion problems
Sebastian 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…
Homogenisation of parabolic/hyperbolic media
Sebastian 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 conver…
On a connection between layer-adapted exponentially graded and S-type meshes
Sebastian 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 m…
Numerical methods for changing type systems
Sebastian 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 severa…
Uniform Error Estimation for Convection-Diffusion Problems
Sebastian Franz
Let us consider the singularly perturbed model problem with homogeneous Dirichlet boundary conditions on on the unit-square…
Superconvergence of a Galerkin FEM for Higher-Order Elements in Convection-Diffusion Problems
S. Franz, H. -G. Roos
In this paper we present a first supercloseness analysis for higher-order Galerkin FEM applied to a singularly perturbed convection-diffusion problem. Using a solution decompositio…
Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems
Sebastian Franz
Considering a singularly perturbed convection-diffusion problem, we present an analysis for a superconvergence result using pointwise interpolation of Gauß-Lobatto type for higher-…
On the sharpness of Green's function estimates for a convection-diffusion problem
Sebastian 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…