Wave number-Explicit Analysis for Galerkin Discretizations of Lossy Helmholtz Problems

We present a stability and convergence theory for the lossy Helmholtz equation and its Galerkin discretization. The boundary conditions are of Robin type. All estimates are explicit with respect to the real and imaginary part of the complex wave number $\zeta\in\mathbb{C}$, $\operatorname{Re}\zeta\geq0$, $\left\vert \zeta\right\vert \geq1$. For the extreme cases $\zeta \in\operatorname*{i}\mathbb{R}$ and $\zeta\in\mathbb{R}_{\geq0}$, the estimates coincide with the existing estimates in the literature and exhibit a seamless transition between these cases in the right complex half plane.Comment: 29 pages, 1 figur

A posteriori error analysis for elliptic pdes on domains with complicated structures

Summary.: The discretisation of boundary value problems on complicated domains cannot resolve all geometric details such as small holes or pores. The model problem of this paper consists of a triangulated polygonal domain with holes of a size of the mesh-width at most and mixed boundary conditions for the Poisson equation. Reliable and efficient a posteriori error estimates are presented for a fully numerical discretisation with conforming piecewise affine finite elements. Emphasis is on technical difficulties with the numerical approximation of the domain and their influence on the constants in the reliability and efficiency estimate

Generalized convolution quadrature with variable time stepping

In this paper, we will present a generalized convolution quadrature for solving linear parabolic and hyperbolic evolution equations. The original convolution quadrature method by Lubich works very nicely for equidistant time steps while the generalization of the method and its analysis to nonuniform time stepping is by no means obvious. We will introduce the generalized convolution quadrature allowing for variable time steps and develop a theory for its error analysis. This method opens the door for further development towards adaptive time stepping for evolution equations. As the main application of our new theory, we will consider the wave equation in exterior domains which is formulated as a retarded boundary integral equatio

A Stable Boundary Integral Formulation of an Acoustic Wave Transmission Problem with Mixed Boundary Conditions

In this paper, we consider an acoustic wave transmission problem with mixed boundary conditions of Dirichlet, Neumann, and impedance type. The transmission interfaces may join the domain boundary in a general way independent of the location of the boundary conditions. We will derive a formulation as a \textit{direct}, \textit{space-time retarded boundary integral equation}, where both Cauchy data are kept as unknowns on the impedance part of the boundary. This requires the definition of single-trace spaces which incorporate homogeneous Dirichlet and Neumann conditions on the corresponding parts on the boundary. We prove the continuity and coercivity of the formulation by employing the technique of operational calculus in the Laplace domain.Comment: 15 pages, 1 figur

Convolution quadrature for the wave equation with impedance boundary conditions

We consider the numerical solution of the wave equation with impedance boundary conditions and start from a boundary integral formulation for its discretization. We develop the generalized convolution quadrature (gCQ) to solve the arising acoustic retarded potential integral equation for this impedance problem. For the special case of scattering from a spherical object, we derive representations of analytic solutions which allow to investigate the effect of the impedance coefficient on the acoustic pressure analytically. We have performed systematic numerical experiments to study the convergence rates as well as the sensitivity of the acoustic pressure from the impedance coefficients. Finally, we apply this method to simulate the acoustic pressure in a building with a fairly complicated geometry and to study the influence of the impedance coefficient also in this situation

Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains

We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional Laplacian. In particular, we develop techniques for the treatment of the dense stiffness matrix including the computation of the entries, the efficient assembly and storage of a sparse approximation and the efficient solution of the resulting equations. The main idea consists of generalising proven techniques for the treatment of boundary integral equations to general fractional orders. Importantly, the approximation does not make any strong assumptions on the shape of the underlying domain and does not rely on any special structure of the matrix that could be exploited by fast transforms. We demonstrate the flexibility and performance of this approach in a couple of two-dimensional numerical examples

Liver transplantation in patients with post-hepatectomy liver failure - A Northern European multicenter cohort study

Background: Liver transplantation (LTX) has been described as a rescue treatment option in severe, intractable post-hepatectomy liver failure (PHLF), but is not considered to be indicated for this condition by many hepatobiliary and transplant surgeons. In this article we describe the clinical experience of five northern European tertiary centers in using LTX to treat selected patients with severe PHLF. Methods: All patients subjected to LTX due to PHLF at the participating centers were identified from prospective clinical databases. Preoperative variables, surgical outcome (both resection surgery and LTX) and follow-up data were assessed.Results: A total of 10 patients treated with LTX due to severe PHLF from September 2008 to May 2020 were identified and included in the study. All patients but one were male and the median age was 70 years (range 49-72). In all patients the indication for liver resection was suspected malignancy, but in six patients post-resection pathology revealed benign or pre-malignant disease. There was no 90-day mortality after LTX. Patients were followed for a median of 49 months (13-153) and eight patients were alive without recurrence at last follow-up.Discussion: In selected patients with PHLF LTX can be a life-saving procedure with low short-term risk.Peer reviewe

A Mass Spectrometry-Based Profiling of Interactomes of Viral DDB1- and Cullin Ubiquitin Ligase-Binding Proteins Reveals NF-κB Inhibitory Activity of the HIV-2-Encoded Vpx

Viruses and hosts are situated in a molecular arms race. To avoid morbidity and mortality, hosts evolved antiviral restriction factors. These restriction factors exert selection pressure on the viruses and drive viral evolution toward increasingly efficient immune antagonists. Numerous viruses exploit cellular DNA damage-binding protein 1 (DDB1)-containing Cullin RocA ubiquitin ligases (CRLs) to induce the ubiquitination and subsequent proteasomal degradation of antiviral factors expressed by their hosts. To establish a comprehensive understanding of the underlying protein interaction networks, we performed immuno-affinity precipitations for a panel of DDB1-interacting proteins derived from viruses such as mouse cytomegalovirus (MCMV, Murid herpesvirus [MuHV] 1), rat cytomegalovirus Maastricht MuHV2, rat cytomegalovirus English MuHV8, human cytomegalovirus (HCMV), hepatitis B virus (HBV), and human immunodeficiency virus (HIV). Cellular interaction partners were identified and quantified by mass spectrometry (MS) and validated by classical biochemistry. The comparative approach enabled us to separate unspecific interactions from specific binding partners and revealed remarkable differences in the strength of interaction with DDB1. Our analysis confirmed several previously described interactions like the interaction of the MCMV-encoded interferon antagonist pM27 with STAT2. We extended known interactions to paralogous proteins like the interaction of the HBV-encoded HBx with different Spindlin proteins and documented interactions for the first time, which explain functional data like the interaction of the HIV-2-encoded Vpr with Bax. Additionally, several novel interactions were identified, such as the association of the HIV-2-encoded Vpx with the transcription factor RelA (also called p65). For the latter interaction, we documented a functional relevance in antagonizing NF-κB-driven gene expression. The mutation of the DDB1 binding interface of Vpx significantly impaired NF-κB inhibition, indicating that Vpx counteracts NF-κB signaling by a DDB1- and CRL-dependent mechanism. In summary, our findings improve the understanding of how viral pathogens hijack cellular DDB1 and CRLs to ensure efficient replication despite the expression of host restriction factors