On the Trace Operator for Functions of Bounded A\mathbb{A}-Variation

In this paper, we consider the space $\mathrm{BV}^{\mathbb A}(\Omega)$ of functions of bounded $\mathbb A$-variation. For a given first order linear homogeneous differential operator with constant coefficients $\mathbb A$, this is the space of $L^1$--functions $u:\Omega\rightarrow\mathbb R^N$ such that the distributional differential expression $\mathbb A u$ is a finite (vectorial) Radon measure. We show that for Lipschitz domains $\Omega\subset\mathbb R^{n}$, $\mathrm{BV}^{\mathbb A}(\Omega)$-functions have an $L^1(\partial\Omega)$-trace if and only if $\mathbb A$ is $\mathbb C$-elliptic (or, equivalently, if the kernel of $\mathbb A$ is finite dimensional). The existence of an $L^1(\partial\Omega)$-trace was previously only known for the special cases that $\mathbb A u$ coincides either with the full or the symmetric gradient of the function $u$ (and hence covered the special cases $\mathrm{BV}$ or $\mathrm{BD}$). As a main novelty, we do not use the fundamental theorem of calculus to construct the trace operator (an approach which is only available in the $\mathrm{BV}$- and $\mathrm{BD}$-setting) but rather compare projections onto the nullspace as we approach the boundary. As a sample application, we study the Dirichlet problem for quasiconvex variational functionals with linear growth depending on $\mathbb A u$

Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology

We develop the analysis of finite element approximations of implicit power-law-like models for viscous incompressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multi–valued, maximal monotone $r$-graph, with $1 < r < \infty$. Using a variety of weak compactness techniques, including Chacon’s biting lemma and Young measures, we show that a subsequence of the sequence of finite element solutions converges to a weak solution of the problem as the finite element discretization parameter $h$ tends to 0. A key new technical tool in our analysis is a finite element counterpart of the Acerbi–Fusco Lipschitz truncation of Sobolev functions

Convergence Analysis for a Finite Element Approximation of a Steady Model for Electrorheological Fluids

In this paper we study the finite element approximation of systems of $p(\cdot)$-Stokes type, where $p(\cdot)$ is a (non constant) given function of the space variables. We derive --in some cases optimal-- error estimates for finite element approximation of the velocity and of the pressure, in a suitable functional setting

Unconditional stability of semi-implicit discretizations of singular flows

A popular and efficient discretization of evolutions involving the singular $p$-Laplace operator is based on a factorization of the differential operator into a linear part which is treated implicitly and a regularized singular factor which is treated explicitly. It is shown that an unconditional energy stability property for this semi-implicit time stepping strategy holds. Related error estimates depend critically on a required regularization parameter. Numerical experiments reveal reduced experimental convergence rates for smaller regularization parameters and thereby confirm that this dependence cannot be avoided in general.Comment: 21 pages, 8 figure