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We examine robustness of exponential dichotomies of boundary value problems for general linear first-order one-dimensional hyperbolic systems. The boundary conditions are supposed to be of types ensuring smoothing solutions in finite time, which includes reflection boundary conditions. We show that the dichotomy survives in the space of continuous functions under small perturbations of all coefficients in the differential equations.
We give an exposition of recent results on regularity and Fredholm properties for first-order one-dimensional hyperbolic PDEs. We show that large classes of boundary operators cause an effect that smoothness increases with time. This property is the key in finding regularizers
(parametrices) for hyperbolic problems. We construct regularizers for periodic problems for dissipative first-order linear hyperbolic PDEs and show that these problems are modeled by Fredholm operators of index zero.
We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic $\lambda$-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory.
Grad-div stabilization has been proved to be a very useful tool in discretizations
of incompressible flow problems. Standard error analysis for inf-sup stable conforming pairs of
finite element spaces predicts that the stabilization parameter should be optimally chosen
to be $\mathcal O(1)$. This paper revisits this choice for the Stokes equations on the basis
of minimizing the $H^1(\Omega)$ error of the velocity and the $L^2(\Omega)$ error of the pressure.
It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle
than known so far in the literature. It depends on the used norm,
the solution, the family of finite
element spaces, and the type of mesh. Depending on the situation, the
optimal
stabilization parameter might range from being very small to very large.
The analytic results
are supported by numerical examples.
Complete damage in linear elastic materials — Modeling, weak formulation and existence results
(2013)
In this work, we introduce a degenerating PDE system with a time-depending
domain for complete damage processes under time-varying
Dirichlet boundary conditions. The evolution of the system is
described by a doubly nonlinear differential inclusion for the damage
process and a degenerating quasi-static balance equation for the displacement field
which are strongly nonlinearly coupled.
In our proposed model, the material
may completely disintegrate which is indispensable for a realistic modeling of
damage processes in elastic materials. Complete damage theories
lead to several mathematical problems since, for instance, coercivity properties
of the free energy are lost and, therefore, several difficulties arise.
For the introduced complete damage model, we propose a classical
formulation and a corresponding suitable weak formulation in an
$SBV$-framework. The main aim is to prove existence of weak solutions
for the introduced degenerating model. In addition, we show that the classical
differential inclusion can be regained from the notion of weak solutions under
certain regularity assumptions which is a novelty in the theory of complete damage
models of this type.
For the existence results, we had to handle the following problem:
During the damage process it might occur that not completely damaged material regions are isolated
from the Dirichlet boundary. In this case, the
deformation field cannot be controlled in the transition from incomplete
to complete damage. To tackle this problem, we consider the evolution
process on a time-depending domain. In this context, two major challenges
arise:
Firstly, the time-dependent domain approach leads to jumps in the energy
which have to be accounted for in the energy inequality of the notion of
weak solutions. To handle this problem, several energy estimates are established
by $\Gamma$-convergence techniques. Secondly, the time-depending domain
might have bad smoothness properties such that Korn's inequality cannot be
applied. To this end, a covering result for such sets with smooth
compactly embedded domains has been shown.
This paper discusses adaptive finite element methods (AFEMs) for the solution of elliptic eigenvalue problems associated with partial differential operators. An adaptive method based on nodal-patch refinement leads to an asymptotic error reduction property for the computed sequence of simple eigenvalues and eigenfunctions. This justifies the use of the proven saturation property for a class of reliable and efficient hierarchical a posteriori error estimators. Numerical experiments confirm that the saturation property is present even for very coarse meshes for many examples; in other cases the smallness assumption on the initial mesh may be severe.
Flux variability analysis (FVA) is an important tool to further analyze the results obtained by flux balance analysis (FBA) on genome-scale metabolic networks. Standard FVA may predict unbounded fluxes through some reactions in the network even if the nutrient uptake rate is bounded. These fluxes violate the second law of thermodynamics. They may be eliminated by extending flux variability analysis with thermodynamic constraints.
We present a new algorithm for efficient flux variability (and flux balance) analysis with thermodynamic constraints, suitable for analyzing genome-scale metabolic networks. We first show that flux balance analysis with thermodynamic constraints is NP-hard. Then we derive a theoretical tractability result, which can be applied to metabolic networks in practice. We use this result to develop a new constraint programming algorithm Fast-tFVA for fast flux variability analysis with thermodynamic constraints (tFVA). Computational comparisons with previous methods demonstrate the efficiency of the new method. For tFVA, a speed-up of factor 30-300 is achieved.
In an analysis of genome-scale metabolic networks in the BioModels database, we found that in 485 out of 716 networks additional irreversible or fixed reactions could be detected.
We discuss the possibility of computing eigenpairs of some prototypical linear second-order self-adjoint elliptic partial differential operator (or its high-resolution finite element discretization) by numerical upscaling techniques. We compute a low-dimensional generalized finite element space that preserves small eigenvalues in a superconvergent way. The approximate eigenpairs are then obtained by solving the corresponding low-dimensional algebraic eigenvalue problem. The rigorous error bounds are based on two-scale decompositions of H1 by means of a certain Clement-type quasi-interpolation operator.
We present a discretization for dynamic large deformation contact problems without friction. Our model is based on Hamilton’s principle, which avoids the explicit appearance of the contact forces. The resulting differential inclusion is discretized in time using a modified midpoint rule. This modification, which concerns the evaluation of the generalized gradient, allows to achieve energy dissipativity. For the space discretization we use a dual-basis mortar method. The resulting spatial algebraic
problems are nonconvex minimization problems with nonconvex inequality constraints. These can be solved efficiently using a trust-region SQP framework with a monotone multigrid inner solver.
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coeffcient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear finescale computations in small patches that have a diameter of order H |log(H)| where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coeffcients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh
size. To solve the arising equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.