arXiv subjects

Sebastian Franz

Publications and source records attributed to Sebastian Franz.

At least 19 recordsLinked to original sources

arXiv2026

Well-posedness and time stepping adaptivity for a class of collocation discretisations of time-fractional subdiffusion equations

Time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$ are discretised in time using collocation methods, which assume that the Caputo derivative of the computed solution is piecewise-polynomial. For such discretisations of any order $m\ge 0$, with any choice of collocation points, we give sufficient conditions for existence and uniqueness of collocation solutions. Furthermore, we investigate the applicability and performance of such schemes in the context of the a-posteriori error estimation and adaptive time stepping algorithms.

math.NA↗

arXiv2026

Numerical Treatment of Non-local Integral Operators in the Framework of Evolutionary Equations

Using the theory of evolutionary equations, we consider abstract differential equations including non-local integral operators. After providing a condition for the well-posedness of the addressed equation we consider a numerical method of approximating its solution. We provide convergence proofs under conditions on the kernel of the integral operator and the solution and finish the paper with some simulation results.

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arXiv2024

Homogenisation for Maxwell and Friends

We refine the understanding of continuous dependence on coefficients of solution operators under the nonlocal $H$-topology viz Schur topology in the setting of evolutionary equations in the sense of Picard. We show that certain components of the solution operators converge strongly. The weak convergence behaviour known from homogenisation problems for ordinary differential equations is recovered on the other solution operator components. The results are underpinned by a rich class of examples that, in turn, are also treated numerically, suggesting a certain sharpness of the theoretical findings. Analytic treatment of an example that proves this sharpness is provided too. Even though all the considered examples contain local coefficients, the main theorems and structural insights are of operator-theoretic nature and, thus, also applicable to nonlocal coefficients. The main advantage of the problem class considered is that they contain mixtures of type, potentially highly oscillating between different types of PDEs; a prototype can be found in Maxwell's equations highly oscillating between the classical equations and corresponding eddy current approximations.

math.AP↗

arXiv2024

On the solution existence for collocation discretizations of time-fractional subdiffusion equations

Time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$ are discretized in time using continuous collocation methods. For such discretizations, we give sufficient conditions for existence and uniqueness of their solutions. Two approaches are explored: the Lax-Milgram Theorem and the eigenfunction expansion. The resulting sufficient conditions, which involve certain $m\times m$ matrices (where $m$ is the order of the collocation scheme), are verified both analytically, for all $m\ge 1$ and all sets of collocation points, and computationally, for all $ m\le 20$. The semilinear case is also addressed.

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arXiv2024

Time stepping adaptation for subdiffusion problems with non-smooth right-hand sides

We consider a time-fractional subdiffusion equation with a Caputo derivative in time, a general second-order elliptic spatial operator, and a right-hand side that is non-smooth in time. The presence of the latter may lead to locking problems in our time stepping procedure recently introduced in [2,4]. Hence, a generalized version of the residual barrier is proposed to rectify the issue. We also consider related alternatives to this generalized algorithm, and, furthermore, show that this new residual barrier may be useful in the case of a negative reaction coefficient.

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arXiv2023

Structure preserving discontinuous Galerkin approximation of a hyperbolic-parabolic system

We study the numerical approximation of a coupled hyperbolic-parabolic system by a family of discontinuous Galerkin space-time finite element methods. The model is rewritten as a first-order evolutionary problem that is treated by the unified abstract solution theory of R. Picard. For the discretization in space, generalizations of the distribution gradient and divergence operators on broken polynomial spaces are defined. Since their skew-selfadjointness is perturbed by boundary surface integrals, adjustments are introduced such that the skew-selfadjointness of the first-order differential operator in space is recovered. Well-posedness of the fully discrete problem and error estimates for the discontinuous Galerkin approximation in space and time are proved.

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arXiv2023

Numerical analysis of a singularly perturbed 4th order problem with a shift term

We consider a one-dimensional singularly perturbed 4th order problem with the additional feature of a shift term. An expansion into a smooth term, boundary layers and an inner layer yields a formal solution decomposition, and together with a stability result we have estimates for the subsequent numerical analysis. With classical layer adapted meshes we present a numerical method, that achieves supercloseness and optimal convergence orders in the associated energy norm. We also consider coarser meshes in view of the weak layers. Some numerical examples conclude the paper and support the theory.

math.NA↗

arXiv2023

Post-processing and improved error estimates of numerical methods for evolutionary systems

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 theory in exponentially weighted spaces which can be exploited in the analysis of numerical methods. The numerical method considered in this paper is a discontinuous Galerkin method in time combined with a conforming Galerkin method in space. Building on our recent paper, we improve some of the results, study the dependence of the numerical solution on the weight-parameter, consider a reformulation and post-processing of its numerical solution. As a by-product we provide error estimates for the dG-C0 method. Numerical simulations support the theoretical findings.

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arXiv2023

A convection-diffusion problem with a large shift on Duran meshes

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 coarser Duran type meshes in places where weak layers appear, is provided. The theoretical results are confirmed by numerical experiments.

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arXiv2022

Green's function estimates for a 2d singularly perturbed convection-diffusion problem: extended analysis

This paper presents an extended version of the article [Franz, S., Kopteva, N.: J. Differential Equations, 252 (2012)]. The main improvement compared to the latter is in that here we additionally estimate the mixed second-order derivative of the Green's function. The case of Neumann conditions along the characteristic boundaries is also addressed. A singularly perturbed convection-diffusion problem is posed in the unit square with a horizontal convective direction. Its solutions exhibit parabolic and exponential boundary layers. Sharp estimates of the Green's function and its first- and second-order derivatives are derived in the $L_1$ norm. The dependence of these estimates on the small diffusion parameter is shown explicitly. The obtained estimates will be used in a forthcoming numerical analysis of the considered problem.

math.AP↗

arXiv2022

Pointwise-in-time a posteriori error control for higher-order discretizations of time-fractional parabolic equations

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 estimation and adaptive time stepping proposed in [7]. We improve the earlier time stepping algorithm based on this theory, and specifically address its stable and efficient implementation in the context of high-order methods. The considered methods include an L1-2 method and continuous collocation methods of arbitrary order, for which adaptive temporal meshes are shown to yield optimal convergence rates in the presence of solution singularities.

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arXiv2022

Numerical analysis of a singularly perturbed convection diffusion problem with shift in space

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 result. Based upon this we provide a numerical analysis of high order finite element method on layer adapted meshes. We also apply a new idea of using a coarser mesh in places where weak layers appear. Numerical experiments confirm our theoretical results.

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arXiv2022

Maximum norm a posteriori error estimates for convection-diffusion problems

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 analysis in the energy norm by Verf\"{u}rth indicates that a dual norm of the {convective derivative of the} error must be added to the natural energy norm in order for the natural residual estimator to be reliable and efficient. We show that the situation is similar for the maximum norm. In particular, we define a mesh-dependent weighted seminorm of the convective error which functions as a maximum-norm counterpart to the dual norm used in the energy norm setting. The total error is then defined as the sum of this seminorm, the maximum norm of the error, and data oscillation. The natural maximum norm residual error estimator is shown to be equivalent to this total error notion, with constant independent of singular perturbation parameters. These estimates are proved under the assumption that certain natural estimates hold for the Green's function for the problem at hand. Numerical experiments confirm that our estimators effectively capture the maximum-norm error behavior for singularly perturbed problems, and can effectively drive adaptive refinement in order to capture layer phenomena.

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arXiv2022

A balanced norm error estimation for the time-dependent reaction-diffusion problem with shift in space

We consider a singularly perturbed time-dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Dur\'{a}n- and S-type we derive a-priori error estimates for the stationary problem. Using a discontinuous Galerkin method in time we obtain error estimates for the full discretisation. Introduction of a weighted scalar products and norms allows us to estimate the error of the time-dependent problem in energy and balanced norm. So far it was open to prove such a result. Error estimates in the energy norm is for the standard finite element discretization in space, and for the error estimate in the balanced norm the computation of the numerical solution is changed by using a different bilinear form. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution.

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arXiv2021

Singularly perturbed reaction-diffusion problems as first order systems

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 $H_{div}$-conforming elements for the second component we provide a convergence analysis on layer adapted meshes and an optimal convergence order in a balanced norm that is comparable with a balanced $H^2$-norm for the second order formulation.

math.NA↗

arXiv2021

Anisotropic $H_{div}$-norm error estimates for rectangular $H_{div}$-elements

For the discretisation of $H_{div}$-functions on rectangular meshes there are at least three families of elements, namely Raviart-Thomas-, Brezzi-Douglas-Marini- and Arnold-Boffi-Falk-elements. In order to prove convergence of a numerical method using them, sharp interpolation error estimates are important. We provide them here in an anisotropic setting for the $H_{div}$-norm.

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arXiv2019

Continuous time integration for changing type systems

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 in a cGP-like norm and conclude the paper with some numerical examples and conclusions.

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arXiv2019

Error estimates in balanced norms of finite element methods for higher order reaction-diffusion problems

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.

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