Multigrid methods for obstacle problems

In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set

On preconditioned Uzawa-type iterations for a saddle point problem with inequality constraints

We consider preconditioned Uzawa iterations for a saddle point Problem with inequality constraints as arising from an implicit time discretization of the Cahn-Hilliard equation with obstacle potential. We present a new class of preconditioners based on linear Schur complements associated with successive approximations of the coincidence set. In numerical experiments, we found superlinear convergence and finite termination

Nonsmooth Newton methods for set-valued saddle point problems

We present a new class of iterative schemes for large scale set-valued saddle point problems as arising, e.g., from optimization problems in the presence of linear and inequality constraints. Our algorithms can be regarded either as nonsmooth Newton-type methods for the nonlinear Schur complement or as Uzawa-type iterations with active set preconditioners. Numerical experiments with a control constrained optimal control problem and a discretized Cahn–Hilliard equation with obstacle potential illustrate the reliability and efficiency of the new approach

On preconditioned Uzawa-type iterations for a saddle point problem with inequality constraints

We consider preconditioned Uzawa iterations for a saddle point problem with inequality constraints as arising from an implicit time discretization of the Cahn-Hilliard equation with an obstacle potential. We present a new class of preconditioners based on linear Schur complements associated with successive approximations of the coincidence set. In numerical experiments, we found superlinear convergence and finite termination

Multilevel Monte Carlo finite element methods for stochastic elliptic variational inequalities

Multilevel Monte Carlo finite element methods (MLMC-FEMs) for the solution of stochastic elliptic variational inequalities are introduced, analyzed, and numerically investigated. Under suitable assumptions on the random diffusion coefficient, the random forcing function, and the deterministic obstacle, we prove existence and uniqueness of solutions of “pathwise” weak formulations. Suitable regularity results for deterministic, elliptic obstacle problems lead to uniform pathwise error bounds, providing optimal-order error estimates of the statistical error and upper bounds for the corresponding computational cost for the classical MC method and novel MLMC-FEMs. Utilizing suitable multigrid solvers for the occurring sample problems, in two space dimensions MLMC-FEMs then provide numerical approximations of the expectation of the random solution with the same order of efficiency as for a corresponding deterministic problem, up to logarithmic terms. Our theoretical findings are illustrated by numerical experiments

Nonsmooth Schur-Newton methods for multicomponent Cahn-Hilliard systems

We present globally convergent nonsmooth Schur–Newton methods for the solution of discrete multicomponent Cahn–Hilliard systems with logarithmic and obstacle potentials. The method solves the nonlinear set-valued saddle-point problems arising from discretization by implicit Euler methods in time and first-order finite elements in space without regularization. Efficiency and robustness of the convergence speed for vanishing temperature is illustrated by numerical experiments

Truncated nonsmooth Newton multigrid methods for convex minimization problems

We present a new inexact nonsmooth Newton method for the solution of convex minimization problems with piecewise smooth, pointwise nonlinearities. The algorithm consists of a nonlinear smoothing step on the fine level and a linear coarse correction. Suitable postprocessing guarantees global convergence even in the case of a single multigrid step for each linear subproblem. Numerical examples show that the overall efficiency is comparable to multigrid for similar linear problems

Numerical simulation of coarsening in binary solder alloys

Coarsening in solder alloys is a widely accepted indicator for possible failure of joints in electronic devices. Based on the well-established Cahn–Larché model with logarithmic chemical energy density (Dreyer and Müller, 2001) [20], we present a computational framework for the efficient and reliable simulation of coarsening in binary alloys. Main features are adaptive mesh refinement based on hierarchical error estimates, fast and reliable algebraic solution by multigrid and Schur–Newton multigrid methods, and the quantification of the coarsening speed by the temporal growth of mean phase radii. We provide a detailed description and a numerical assessment of the algorithm and its different components, together with a practical application to a eutectic AgCu brazing alloy

A nonsmooth Newton multigrid method for a hybrid, shallow model of marine ice sheets

The time evolution of ice sheets and ice shelves is model by combining a shallow lubrication approximation for shear deformation with the shallow shelf approximation for basal sliding, along with the mass conservation principle. At each time step two p-Laplace problems and one transport problem are solved. Both p-Laplace problems are formulated as minimisation problems. They are approximated by a finite element truncated nonsmooth Newton multigrid method. As an illustration, we compute the steady state shape of an idealized ice sheet/shelf system

Hierarchical error estimates for the energy functional in obstacle problems

We present a hierarchical a posteriori error analysis for the minimum value of the energy functional in symmetric obstacle problems. The main result is that the error in the energy minimum is, up to oscillation terms, equivalent to an appropriate hierarchical estimator. The proof does not invoke any saturation assumption. We even show that small oscillation implies a related saturation assumption. In addition, we prove efficiency and reliability of an a posteriori estimate of the discretization error and thereby cast some light on the theoretical understanding of previous hierarchical estimators. Finally, we illustrate our theoretical results by numerical computations