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An adaptive a posteriori error estimator based finite element method for the
numerical solution of a coupled Cahn-Hilliard/Navier-Stokes system with a
double-obstacle homogenous free (interfacial) energy density is proposed. A
semi-implicit Euler scheme for the time-integration is applied which results in
a system coupling a quasi-Stokes or Oseen-type problem for the fluid flow to
a variational inequality for the concentration and the chemical potential according
to the Cahn-Hilliard model [13]. A Moreau-Yosida regularization is
employed which relaxes the constraints contained in the variational inequality
and, thus, enables semi-smooth Newton solvers with locally superlinear convergence
in function space. Moreover, upon discretization this yields a mesh
independent method for a fixed relaxation parameter. For the finite dimensional
approximation of the concentration and the chemical potential piecewise
linear and globally continuous finite elements are used, and for the numerical
approximation of the fluid velocity Taylor-Hood finite elements are employed.
The paper ends by a report on numerical examples showing the efficiency of the
new method
We derive a new representation of Lagrangian subspaces in the form
%
\[
{\mathrm Im}\Pi^T [I,X]^T,
\]
%
where $\Pi$ is a symplectic matrix which is the product of a permutation matrix and a real orthogonal diagonal matrix, and $X$ satisfies
%
\[
\abs{X_{ij}} \leq \begin{cases}1 & \text{if $i=j$,}\\ \sqrt{2} & \text{if $i\neq j$.} \end{cases}
\]
%
This representation allows to limit element growth in the context of doubling algorithms
for the computation of Lagrangian subspaces and the solution of Riccati equations.
It is shown that a simple doubling algorithm using this representation can reach full machine accuracy on a wide range of problems, obtaining invariant subspaces of the same quality as those computed by the state-of-the-art algorithms based on orthogonal transformations.
The same idea carries over to representations of arbitrary subspaces and can be used
for other types of structured pencils.
This note constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying diffusion tensor. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially w.r.t. the number of element layers in the patches. Hence, on a uniform mesh of size H, patches of diameter log(1/H) are sufficient to preserve the convergence rates of the classical P1-FEM for the Poisson problem.
The analysis does not rely on regularity of the solution or scale separation in the coefficient.
The result justifies the use of the class of variational multiscale methods, introduced in [Comput. Methods Appl. Mech. Engrg., 196:2313--2324, 2007].
R is a language and environment for statistical computing and graphics. It can be considered an alternative implementation of the S language developed in the 1970s and 1980s for data analysis and graphics (Becker and Chambers, 1984; Becker et al., 1988). The R language is part of the GNU project and offers versions that compile and run on almost every major operating system currently available. We highlight several R packages built specifically for the analysis of neuroimaging data in the context of functional MRI, diffusion tensor imaging, and dynamic contrast-enhanced MRI. We review their methodology and give an overview of their capabilities for neuroimaging. In addition we summarize some of the current activities in the area of neuroimaging software development in R.
The well-known Kalman-Yakubovich-Popov Lemma establishes an equivalence between dissipativity and the solvability of a linear matrix inequality. In this paper we strengthen this result by showing the equivalence of dissipativity to the solvability of a so-called Lur'e equation, which mainly is a linear matrix inequality with a rank minimizing condition. Finally, we apply the result to standard systems to obtain the well-known result about the solvability of the algebraic Riccati equation.
We investigate geodesic finite elements for functions with values in a space of zero curvature, like a torus or the M\"obius strip. Unlike in the general case, a closed-form expression for geodesic finite element functions is then available. This simplifies computations, and allows us to prove optimal estimates for the interpolation error in 1d and 2d. We also show the somewhat surprising result that the discretization by Kirchhoff transformation of the Richards equation proposed by Berninger et al. is a discretization by geodesic finite elements in the manifold $\mathbb{R}$ with a special metric.
We introduce functional perturbation results for PDE eigenvalue problems including the functional backward error
and the functional condition number. These results are used to establish a combined a posteriori error estimator embodying the discretization and the approximation error for the simple eigenpair.
Based on known perturbation results in $H^{1}(\Omega)$ and $H^{-1}(\Omega)$ norms and a standard residual a posteriori error estimator, a balancing AFEM algorithm is proposed. The stopping criterion for the eigensolver is based on the equilibrating strategy, i.e., iterations proceed as long as the discrete part of the error estimator dominates the continuous part. All our statements are illustrated with several numerical examples.
We consider anisotropic Allen--Cahn equations with interfacial energy
induced by an anisotropic surface energy density $\gamma$.
Assuming that $\gamma$
is positive, positively homogeneous of degree one,
strictly convex in tangential directions to the unit sphere,
and sufficiently smooth, we show stability of
various time discretizations. In particular,
we consider a fully implicit and a linearized time discretization
of the interfacial energy combined with implicit
and semi-implicit time discretizations
of the double-well potential. In the semi-implicit variant,
concave terms are taken explicitly.
The arising discrete spatial problems are solved by
globally convergent truncated nonsmooth Newton multigrid methods.
Numerical experiments show the accuracy of the different
discretizations.
We also illustrate that pinch-off under anisotropic
mean curvature flow is no longer frame invariant,
but depends on the orientation of the initial configuration.
This paper deals with a three-dimensional mixture model describing materials undergoing phase transition with thermal expansion. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. A global solution for this thermodynamically consistent problem is obtained by using a fixed-point argument combined with global energy estimates.
We consider three-dimensional models for rate-independent processes describing materials undergoing phase transformations with heat transfer. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. Under appropriate regularity assumptions on the initial data, we prove the existence a global solution for this thermodynamically consistent system, by using a fixed-point argument combined with global energy estimates.
We consider rate-independent evolutionary systems over a physically domain Ω that are governed by simple hysteresis operators at each material point. For multiscale systems where ε denotes the ratio between the microscopic and the macroscopic length scale, we show that in the limit ε → 0 we are led to systems where the hysteresis operators at each macroscopic point is a generalized Prandtl-Ishlinskii operator.
The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs-Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end, approximate solutions are constructed by means of variational problems
for energy functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law in a weak generalized BV-formulation.
The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and de- velopment of numerical methods. Though much can be said for broad classes of elliptic MPECs in both polyhedric and non-polyhedric settings, the calculation becomes significantly more com- plicated when additional constraints are imposed on the control. In this paper we develop three derivation methods for constrained MPEC problems: via concepts from variational analysis, via penalization of the control constraints, and via penalization of the lower-level problem with the subsequent regularization of the resulting nonsmoothness. The developed methods and obtained results are then compared and contrasted.
We analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps.
Hence, we consider rate-independent damage models as limits of systems driven by viscous, rate-dependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arc-length reparameterization. In this way, in the limit we obtain a novel formulation for the rate-independent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.
We consider linear differential-algebraic m-input m-output systems with positive
strict relative degree or proper inverse transfer function; in the single-input single-output case these
two disjoint classes make the whole of all linear DAEs without feedthrough term. Structural properties
- such as normal forms (i.e. the counterpart to the Byrnes-Isidori form for ODE systems), zero
dynamics, and high-gain stabilizability - are analyzed for two purposes: first, to gain insight into the
system classes and secondly, to solve the output regulation problem by funnel control. The funnel
controller achieves tracking of a class of reference signals within a pre-specified funnel; this means in
particular, the transient behaviour of the output error can be specified and the funnel controller does
neither incorporate any internal model for the reference signals nor any identification mechanism, it
is simple in its design. The results are illuminated by position and velocity control of a mechanical
system encompassing springs, masses, and dampers.
A Composite Finite Element Method approximates linear elliptic boundary value problems of Dirichlet type with discontinuous coefficients at possibly high contrast. The challenge is the discontinuity in the coefficient across some interface which is not necessarily resolved by the underlying finite element mesh. The method is non-conforming in the sense that shape functions preserve continuity across the interface only in an approximative way. However, the method allows to balance the non-conformity and the best approximation error in such a way that the total discretization error is optimal with regard to the mesh size and independent of contrast.
This paper presents some weighted H2-regularity estimates for a model Poisson problem with discontinuous coefficient at high contrast. The coefficient represents a random particle reinforced composite material, i.e., highly conducting circular particles are randomly distributed in some background material with low conductivity. Based on these regularity results we study the percolation of thermal conductivity of the material as the volume fraction of the particles is close to the jammed state. We proof that the characteristic percolation behavior of the material is well captured by standard conforming finite element models.
Discrete Laplace--Beltrami operators on polyhedral surfaces play an important role for various applications in geometry processing and related areas like physical simulation or computer graphics. While discretizations of the weak Laplace--Beltrami operator are well-studied, less is known about the strong form. We present a principle for constructing strongly consistent discrete Laplace--Beltrami operators based on the cotan weights. The consistency order we obtain, improves previous results reported for the mesh Laplacian. Furthermore, we prove consistency of the discrete Willmore energies corresponding to the discrete Laplace--Beltrami operators.
In this paper a high-order finite element method with curvilinear elements is proposed for the simulation of plasmonic structures. Most finite element packages use low order basis functions and non-curved elements, which is very costly for demanding problems such as the simulation of nano-antennas. To enhance the performance of finite elements, we use curvilinear quadrilateral elements to calculate the near-field from an impinging plane wave with
second order absorbing boundary conditions. The magnetic field amplitude on the surface of one object is compared with a computation based on a multiple multipole expansion. Moreover, the convergence behavior of p-FEM with absorbing boundary conditions motivate an adaptive strategy of polynomial degree enhancement and enlargement of the domain.
The focus of this note lies on the numerical analysis of models describing the propagation
of a single crack in a linearly elastic material. The evolution of the crack is modeled as a
rate-independent process based on the Griffith criterion. We follow two different approaches
for setting up mathematically well defined models: the global energetic approach and an
approach based on a viscous regularization.
We prove the convergence of solutions of fully discretized models (i.e. with respect to time
and space) and derive relations between the discretization parameters (mesh size, time step
size, viscosity parameter, crack increment) which guarantee the convergence of the schemes.
Further, convergence rates are provided for the approximation of energy release rates by
certain discrete energy release rates. Thereby we discuss both, models with self-contact
conditions on the crack faces as well as models with pure Neumann conditions on the crack
faces. The convergence proofs rely on regularity estimates for the elastic fields close to the
crack tip and local and global finite element error estimates. Finally the theoretical results
are illustrated with some numerical calculations.