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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.
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.
We present an extension module for the Dune system. This module, called dune-subgrid, allows to mark elements of another Dune hierarchical grid. The set of marked elements can then be accessed as a Dune grid in its own right. dune-subgrid is free software and is available for download. We describe the functionality and use of dune-subgrid, comment on its implementation, and give two example applications.
First, we show how dune-subgrid can be used for micro-FE simulations of trabecular bone. Then we present an algorithm that allows to use exact residuals for the adaptive solution of the spatial problems of time-discretized evolution equations.
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 energy of the exact solution
is, up to data oscillation, equivalent to an appropriate
hierarchical estimator. The proof of the main result does not
invoke any saturation assumption. Moreover, we prove an a
posteriori error estimate indicating that the estimator from
\cite{RHWHoppe_RKornhuber_1994a} is asymptotically reliable and we
give sufficient conditions for the validity of a saturation
assumption. Finally, we corroborate and complement our theoretical
results with numerical experiments.
We introduce nonsmooth Schur-Newton methods for the solution of the nonlinear discrete saddle-point problems arising from discretized vector-valued Cahn-Hilliard equations with logarithmic and obstacle potentials. The discrete problems are obtained by semi-implicit discretization in time and a first order finite element discretization in space. We incorporate the linear constraints that enforce solutions to stay on the Gibbs simplex using Lagrangian multipliers and prove existence of these multipliers under the assumption of a non-trivial initial condition for the order parameters.
Coarsening in solder alloys is a widely accepted indicator for
possible failure of joints in electronic devices.
Based on the well-established Cahn--Larch\'e model
with logarithmic chemical energy density (see, e.g., Dreyer and Mueller 2001),
we present a numerical environment 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.