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

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

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

Optimal and robust a posteriori error estimates in L∞(L2) for the approximation of Allen-Cahn equations past singularities

Optimal a posteriori error estimates in L∞(L2) 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

Mini-Workshop: Numerical Analysis for Non-Smooth PDE-Constrained Optimal Control Problems

This mini-workshop brought together leading experts working on various aspects of numerical analysis for optimal control problems with nonsmoothness. Fifteen extended abstracts summarize the presentations at this mini-workshop

Numerical Methods for Deterministic and Stochastic Phase Field Models of Phase Transition and Related Geometric Flows

This dissertation consists of three integral parts with each part focusing on numerical approximations of several partial differential equations (PDEs). The goals of each part are to design, to analyze and to implement continuous or discontinuous Galerkin finite element methods for the underlying PDE problem. Part One studies discontinuous Galerkin (DG) approximations of two phase field models, namely, the Allen-Cahn and Cahn-Hilliard equations, and their related curvature-driven geometric problems, namely, the mean curvature flow and the Hele-Shaw flow. We derive two discrete spectrum estimates, which play an important role in proving the sharper error estimates which only depend on a negative power of the singular perturbation parameter ε [epsilon] instead of an exponential power. It is also proved that the zero level sets of the numerical solutions of the Allen-Cahn equation and the Cahn-Hilliard equation approximate the mean curvature flow and the Hele-Shaw flow respectively. Numerical experiments are carried out to verify the theoretical results and to compare the zero level sets of the numerical solutions and the geometric motions. Part Two focuses on finite element approximations of stochastic geometric PDEs including the phase field formulation of a stochastic mean curvature flow and the level set formulation of the stochastic mean curvature flow. Both formulations give PDEs with gradient-type multiplicative noises. We establish PDE energy laws and the Hölder [Holder] continuity in time for the exact solutions. Moreover, optimal error estimates are derived, and various numerical experiments are carried out to study the interplay of the geometric evolution and gradient-type noises. Part Three studies finite element methods for a quasi-static model of poroelasticity, which is a fluid-solid interaction multiphysics system at pore scale. We reformulate the original multiphysics system into a new system which explicitly reveals the diffusion process and has a built-in mechanism to overcome the locking phenomenon . Fully discrete finite element methods are proposed for approximating the new system. We derive a discrete energy law and optimal error estimates for our finite element methods. Numerical experiments are also provided to verify the theoretical results and to confirm that the locking phenomenon has indeed been overcome

Mini-Workshop: Adaptive Methods for Control Problems Constrained by Time-Dependent PDEs

Optimization problems constrained by time-dependent PDEs (Partial Differential Equations) are challenging from a computational point of view: even in the simplest case, one needs to solve a system of PDEs coupled globally in time and space for the unknown solutions (the state, the costate and the control of the system). Typical and practically relevant examples are the control of nonlinear heat equations as they appear in laser hardening or the thermic control of flow problems (Boussinesq equations). Specifically for PDEs with a long time horizon, conventional time-stepping methods require an enormous storage of the respective other variables. In contrast, adaptive methods aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities and are, therefore, most promising