Rüdiger Müller

Robust A Priori and A Posteriori Error Analysis for the Approximation of Allen-Cahn and Ginzburg-Landau Equations Past Topological Changes

A priori and a posteriori error estimates are derived for the numerical approximation of scalar and complex valued phase field models. Particular attention is devoted to the dependence of the estimates on a small parameter and to the validity of the estimates in the presence of topological changes in the solution that represents singular points in the evolution. For typical singularities the estimates depend on the inverse of the parameter in a polynomial as opposed to exponential dependence of estimates resulting from a straightforward error analysis. The estimates naturally lead to adaptive mesh refinement and coarsening algorithms. Numerical experiments illustrate the reliability and efficiency of this approach for the evolution of interfaces and vortices that undergo topological changes.

Discontinuous Galerkin Finite Element Convergence for Incompressible Miscible Displacement Problems of Low Regularity

In this article we analyze the numerical approximation of incompressible miscible displacement problems with a combined mixed finite element and discontinuous Galerkin method under minimal regularity assumptions. The main result is that sequences of discrete solutions weakly accumulate at weak solutions of the continuous problem. In order to deal with the nonconformity of the method and to avoid overpenalization of jumps across interelement boundaries, the careful construction of a reflexive subspace of the space of bounded variation, which compactly embeds into

A posteriori error controlled local resolution of evolving interfaces for generalized Cahn–Hilliard equations

L^2(\Omega)

Quasi-optimal and robust a posteriori error estimates in ^{∞}(²) for the approximation of Allen-Cahn equations past singularities

, and of a lifting operator, which is compatible with the nonlinear diffusion coefficient, are required. An equivalent skew-symmetric formulation of the convection and reaction terms of the nonlinear partial differential equation allows us to avoid flux limitation and nonetheless leads to an unconditionally stable and convergent numerical method. Numerical experiments underline the robustness of the proposed algorithm.

Error control for the approximation of Allen–Cahn and Cahn–Hilliard equations with a logarithmic potential

For equations of generalized Cahn–Hilliard type we present an a posteriori error analysis that is robust with respect to a small interface length scale γ . We propose the solution of a fourth order elliptic eigenvalue problem in each time step to gain a fully computable error bound, which only depends polynomially (of low order) on the inverse of γ . A posteriori and a priori error bounds for the eigenvalue problem are also derived. In numerical examples we demonstrate that this approach extends the applicability of robust a posteriori error estimation as it removes restrictive conditions on the initial data. Moreover we show that the computation of the principal eigenvalue allows the detection of critical points during the time evolution that limit the validity of the estimate.

Stable Crank-Nicolson Discretisation for Incompressible Miscible Displacement Problems of Low Regularity

Quasi-optimal a posteriori error estimates in L ∞ (0,T; L 2 (Ω)) are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming, nonconforming, mixed, and discontinuous Galerkin methods. Numerical experiments illustrate the theoretical results.

Error Controlled Local Resolution of Evolving Interfaces for Generalized Cahn-Hilliard Equations

A fully computable upper bound for the finite element approximation error of Allen–Cahn and Cahn–Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.

Die kalte Zunge

In this article we study the numerical approximation of incompressible miscible displacement problems with a linearised Crank–Nicolson time discretisation, combined with a mixed finite element and discontinuous Galerkin method. At the heart of the analysis is the proof of convergence under low regularity requirements. Numerical experiments demonstrate that the proposed method exhibits second-order convergence for smooth and robustness for rough problems.

Modeling of electrochemical double layers in thermodynamic non-equilibrium

For phase field equations of generalized Cahn-Hilliard type, we present an a posteriori error analysis that is robust with respect to a small interface length scale γ which enters the model as a regularizing parameter. By the solution of a fourth order elliptic eigenvalue problem in each time step we gain a fully computable error bound. In accordance with theoretical results, this error bound only depends on the inverse of the small parameter in a low order polynomial for a smooth evolution of the interface. We apply the general framework to the technologically relevant Cahn-Hilliard system coupled with homogeneous elasticity. The derived estimators can be used for adaptive mesh refinement and coarsening. In numerical examples we illustrate that the computation of the principal eigenvalue allows the detection of critical points during the time evolution like merging of interfaces or other topological changes. Moreover, it confirms theoretical predictions about fast relaxation of nonsmooth components in the initial data.

A new perspective on the electron transfer: recovering the Butler–Volmer equation in non-equilibrium thermodynamics

Gefuhlte Temperaturen. Ist ein Null Grad Celsius kalter Metallstab eigentlich kalter als ein Holzstab mit der selben Temperatur? Rein physikalisch gesehen naturlich nicht, aber wenn wir beide Stabe anfassen, kommt uns der Metallstab deutlich kalter vor. Und wer kennt nicht die Szene aus dem Film Dumm und Dummer in der Harry mit seiner Zunge am Metallrahmen des Skilifts hangen bleibt.Wurde das auch passieren, wenn man an einem eiskalten Stuck Holz lecken wurde? Wohl kaum, doch woran liegt das eigentlich? Unterschiedliche Materialien haben verschiedene Fahigkeiten, Warme zu ubertragen und zu leiten. So transportiert Metall die von der Zunge ausgehende Warme sehr schnell weiter und verandert seine Temperatur kaum, wahrend die Zunge abkuhlt. Holz hingegen leitet Warme fast gar nicht und daher wird der Teil, der von der Zunge beruhrt wird, aufgewarmt.