Integral Foliated Simplicial Volume of Aspherical Manifolds

We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated simplicial volume is monotone with respect to weak containment of measure preserving actions and yields upper bounds on (integral) homology growth. Using ergodic theory we prove that simplicial volume, integral foliated simplicial volume and stable integral simplicial volume coincide for closed hyperbolic 3-manifolds and closed aspherical manifolds with amenable residually finite fundamental group (being equal to zero in the latter case). However, we show that integral foliated simplicial volume and the classical simplicial volume do not coincide for hyperbolic manifolds of dimension at least 4

Isometric embeddings in bounded cohomology

This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X,Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X,Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along pi_1-injective boundary components with amenable fundamental group

Stabilized Galerkin for transient advection of differential forms

We deal with the discretization of generalized transient advection problems for differential forms on bounded spatial domains. We pursue an Eulerian method of lines approach with explicit timestepping. Concerning spatial discretization we extend the jump stabilized Galerkin discretization proposed in [H. Heumann and R. Hiptmair, Stabilized Galerkin methods for magnetic advection, Math. Modelling Numer. Analysis, 47 (2013), pp. 1713{ 1732] to forms of any degree and, in particular, advection velocities that may have discontinuities resolved by the mesh. A rigorous a priori convergence theory is established for Lipschitz continuous velocities, conforming meshes and standard nite element spaces of discrete differential forms. However, numerical experiments furnish evidence of the good performance of the new method also in the presence of jumps of the advection velocity

Stabilized Galerkin for Transient Advection of Differential Forms

We deal with the discretization of generalized transient advection problems for differentialforms on bounded spatial domains. We pursue an Eulerian method of lines approachwith explicit time-stepping. Concerning spatial discretization we extend the jump stabilizedGalerkin discretization proposed in [H. Heumann and R. Hiptmair, StabilizedGalerkin methods for magnetic advection, Math. Modelling Numer. Analysis, 47 (2013),pp. 1713–1732] to forms of any degree and, in particular, advection velocities that mayhave discontinuities resolved by the mesh. A rigorous a priori convergence theory is establishedfor Lipschitz continuous velocities, conforming meshes and standard finite elementspaces of discrete differential forms. However, numerical experiments furnish evidence ofthe good performance of the new method also in the presence of jumps of the advectionvelocity

Auxiliary space preconditioners for SIP-DG discretizations of H(curl)-elliptic problems with discontinuous coefficients

We propose a family of preconditioners for linear systems of equations arising from a piecewise polynomial symmetric interior penalty discontinuous Galerkin discretization of H(curl,\Omega)-elliptic boundary value problems on conforming meshes. The design and analysis of the proposed preconditioners rely on the auxiliary space method (ASM) employing an auxiliary space of H(curl,ω)-conforming finite element functions together with a relaxation technique (local smoothing). On simplicial meshes, the proposed preconditioner enjoys asymptotic optimality with respect to mesh refinement. It is also robust with respect to jumps in the coefficients ? and b in the second-and zeroth-order parts of the operator, respectively, except when the problem changes from curl-dominated to reaction-dominated and vice versa. On quadrilateral/hexahedral meshes some of the proposed ASM solvers may fail, since the related H(curl,ω)-conforming finite element space does not provide a spectrally accurate discretization. Extensive numerical experiments are included to verify the theory and assess the performance of the preconditioners

Auxiliary space preconditioners for a DG discretization of H(curl; Ω)-elliptic problem on hexahedral meshes

We present a family of preconditioners based on the auxiliary space method for a discontinuous Galerkin discretization on cubical meshes of H(curl;Ω)-elliptic problems with possibly discontinuous coefficients. We address the influence of possible discontinuities in the coefficients on the asymptotic performance of the proposed solvers and present numerical results in two dimensions