It is well known that the classical local projection method as well as residual-based stabilization techniques, as for instancestreamline upwind Petrov-Galerkin (SUPG), are optimal on isotropicmeshes. Here we extend the local projection stabilization for the Navier-Stokes system to anisotropic quadrilateral meshes in two spatial dimensions. Wedescribe the new methodand prove an a priori error estimate. This method leads on anisotropic meshes to qualitatively better convergence behavior than other isotropicstabilization methods.The capability of the methodis illustrated by means of two numerical test problems.

The Ultra Weak Variational Formulation (UWVF) of the Helmholtz equationprovides a variational framework suitable for discretization using plane wave solutionsof an appropriate adjoint equation. Currently convergence of the method is only proved on the boundary of the domain. However substantial computational evidence exists showing that the method also converges throughout the domain of the Helmholtz equation. In this paper we exploit the fact that the UWVF is essentially an upwind discontinuous Galerkin method to prove convergence of the solution in the special case where there is no absorbing medium present. We also provide some other estimates in the case when absorption is present, and give some simple numerical results to test the estimates. We expect that similar techniques can be used to prove error estimates for the UWVF applied to Maxwell's equations and elasticity.

A class of compatible spatial discretizations for solving partial differential equations is presented. A discrete exact sequence framework is developed to classify these methods which include the mimetic and the covolume methods as well as certain low-order finite element methods. This construction ensures discrete analogs of the differential operators that satisfy the identities and theorems of vector calculus, in particular a Helmholtz decomposition theorem for the discrete function spaces. This paper demonstrates that these methods differ only in their choice of discrete inner product. Finally, certain uniqueness results for the covolume inner product are shown.

We propose a model for a medical device, called a stent, designed for the treatment of cerebral aneurysms. The stent consists of a grid, immersed in the blood flow and located at the inlet of the aneurysm. It aims at promoting a clot within the aneurysm. The blood flow is modelled by the incompressible Navier-Stokes equations and the stent by a dissipative surface term. We propose a stabilized finite element method for this model and we analyse its convergence in the case of the Stokes equations. We present numerical results for academical test cases, and on a realistic aneurysm obtained from medical imaging.

This paper is concerned with numerical methods for compressible multicomponent fluids. The fluid components are assumed immiscible, and areseparated by material interfaces, each endowed with its own equation of state (EOS). Cell averages of computational cells that are occupiedby several fluid components require a “mixed-cell” EOS, which may not always be physically meaningful, and often leads to spurious oscillations. We present a new interface tracking algorithm, which avoids using mixed-cell information by solving the Riemann problembetween its single-fluid neighboring cells. The resulting algorithm is oscillation-free for isolated material interfaces, conservative, andtends to produce almost perfect jumps across material fronts. The computational framework is general and may be used in conjunction withone's favorite finite-volume method. The robustness of the method is illustrated on shock-interface interaction in one space dimension,oscillating bubbles with radial symmetry and shock-bubble interaction in two space dimensions.

In this work, we consider dynamic frictionless contact with adhesionbetween a viscoelastic body of the Kelvin-Voigt type and astationary rigid obstacle, based on the Signorini's contact conditions.Including the adhesion processes modeled by the bonding field, a newversion of energy function is defined. We use the energy functionto derive a new form of energy balance which is supported by numericalresults. Employing the time-discretization, we establish a numerical formulation and investigate the convergence of numerical trajectories. The fullydiscrete approximation which satisfies the complementarity conditionsis computed by using the nonsmooth Newton's method with the Kanzow-Kleinmichelfunction. Numerical simulations of a viscoelastic beam clamped attwo ends are presented.

The main objective of this paper is to provenew necessary conditions to the existence ofKAM tori. To do so, we develop a set ofexplicit a-priori estimates for smoothsolutions of Hamilton-Jacobi equations,using a combination of methods fromviscosity solutions,KAM and Aubry-Mather theories.These estimatesare validin anyspace dimension, and can be checked numericallyto detect gaps between KAM tori and Aubry-Mather sets.We apply these results to detect non-integrable regions in several examples such as a forced pendulum, two coupled penduli, andthe double pendulum.

The incompressible MHD equations couple Navier-Stokes equations with Maxwell's equationsto describe the flow of a viscous, incompressible, and electrically conducting fluid ina Lipschitz domain $\Omega \subset \mathbb{R}^3$ .We verify convergence of iterates of different coupling anddecoupling fully discrete schemes towards weak solutions forvanishing discretization parameters. Optimal first order of convergence is shown in the presence of strong solutions for a splitting scheme which decouplesthe computation of velocity field, pressure, and magnetic fields atevery iteration step.