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- Cahn-Larche system (1)
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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.
Multi-Level Monte-Carlo Finite Element Methods for stochastic elliptic variational inequalities
(2013)
Multi-Level Monte-Carlo Finite Element (MLMC--FE) methods
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 ``mean-square''
and ``pathwise'' 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
classical Monte--Carlo and novel MLMC--FE methods.
Utilizing suitable multigrid solvers for the occurring sample problems,
in two space dimensions
MLMC--FE methods 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.
We consider a non-isothermal multi-phase field model.
We subsequently discretize implicitly in time and with
linear finite elements. The arising algebraic problem is
formulated in two variables where one is the multi-phase
field, and the other contains the inverse temperature field.
We solve this saddle point problem numerically by a
non-smooth Schur-Newton approach using truncated
non-smooth Newton multigrid methods. An application in
grain growth as occurring in liquid phase crystallization
of silicon is considered.
We present a globally convergent method for the solution of frictionless large deformation contact problems involving hyperelastic materials.
For the discretisation we apply the dual mortar method which is known to be more stable than node-to-segment approaches. The resulting non-convex constrained minimisation problems are solved using a filter–trust-region scheme.
This method combines several techniques from non-linear optimisation to achieve global convergence towards first-order optimal points. To speed up the method inexact linearisations of the non-penetration constraint are used
whenever the current iterate is far away from a critical point. A monotone multigrid method is applied for the fast solution of the constrained Newton
problems.