25 research outputs found
A note on Poincar\'e- and Friedrichs-type inequalities
We introduce a simple criterion to check coercivity of bilinear forms on
subspaces of Hilbert-spaces. The presented criterion allows to derive many
standard and non-standard variants of Poincar\'e- and Friedrichs-type
inequalities with very little effort
CRC 1114 - Report Membrane Deformation by N-BAR Proteins: Extraction of membrane geometry and protein diffusion characteristics from MD simulations
We describe simulations of Proteins and artificial pseudo-molecules
interacting and shaping lipid bilayer membranes. We extract protein diffusion
Parameters, membrane deformation profiles and the elastic properties of the
used membrane models in preparation of calculations based on a large scale
continuum model
Nonsmooth Schur-Newton methods for vector-valued Cahn-Hilliard equations
We present globally convergent nonsmooth Schur-Newton methods for the solution
of discrete vector-valued Cahn-Hilliard equations with logarithmic and
obstacle potentials. The method solves the nonlinear set-valued saddle-point
problems as 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 for phase-field brittle-fracture problems
We propose the Truncated Nonsmooth Newton Multigrid Method (TNNMG) as a
solver for the spatial problems of the small-strain brittle-fracture
phase-field equations. TNNMG is a nonsmooth multigrid method that can solve
biconvex, block-separably nonsmooth minimization problems in roughly the time
of solving one linear system of equations. It exploits the variational
structure inherent in the problem, and handles the pointwise irreversibility
constraint on the damage variable directly, without penalization or the
introduction of a local history field. Memory consumption is significantly
lower compared to approaches based on direct solvers. In the paper we introduce
the method and show how it can be applied to several established models of
phase-field brittle fracture. We then prove convergence of the solver to a
solution of the nonsmooth Euler-Lagrange equations of the spatial problem for
any load and initial iterate. Numerical comparisons to an operator-splitting
algorithm show a speed increase of more than one order of magnitude, without
loss of robustness
A Variational Approach to Particles in Lipid Membranes
A variety of models for the membrane-mediated interaction of particles in
lipid membranes, mostly well-established in theoretical physics, is reviewed
from a mathematical perspective. We provide mathematically consistent
formulations in a variational framework, relate apparently different modelling
approaches in terms of successive approximation, and investigate existence and
uniqueness. Numerical computations illustrate that the new variational
formulations are directly accessible to effective numerical methods