In this paper we derive an algorithm for solving the Stokes{equations for the example of the Driven{Cavity{Problem. We treat the Stokes{equations in the divergence{free weak formulation and show the construction of divergence{free reenable functions, which will be used as trial functions in the Galerkin{method. The system will be solved by means of the preconditioned cg{method. We make use of m...

This paper establishes a foundation of non-conforming boundary elements. We present a discrete weak formulation of hypersingular integral operator equations that uses Crouzeix–Raviart elements for the approximation. The cases of closed and open polyhedral surfaces are dealt with. We prove that, for shape regular elements, this non-conforming boundary element method converges and that, up to log...

Classes with few models have 'homogeneous-universal' models This paper is concerned with a class K of models and an abstract notion of submodel ≤. Experience in first order model theory has shown the desirability of finding a 'monster model' to serve as a universal domain for K. In the original constructions of Jónsson and Fraissè, K was a universal class and ordinary substructure played the ro...

The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared. This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at certain wavenumbers (those that correspon...

We consider isentropic gas dynamics equations with unilateral constraint on the density and mass loss. The and pressureless pressure laws are considered. We propose an entropy weak formulation of the system that incorporates the constraint and Lagrange multiplier, for which we prove weak stability and existence of solutions. The nonzero pressure model is approximated by a kinetic BGK relaxation...

This paper deals with the weak formulation of a free moving boundary problem arising in theoretical glaciology Considering shallow ice sheet ow we present the mathematical analysis and the numerical resolution of the second order nonlinear degenerate parabolic equation modelling in the isothermal case the ice sheet non newtoniandynamics An obstacle problem is then deduced and analysed The exist...

Well-posed space-time variational formulations in fractional order Bochner–Sobolev spaces are proposed for parabolic partial differential equations, and in particular for the instationary Stokes and Navier–Stokes equations on bounded Lipschitz domains. The latter formulations include the pressure variable as a primal unknown and so account for the incompressibility constraint via a Lagrange mul...

In this work we are considered with parameter optimization of elliptic multiscale problems with macroscopic optimization functionals and microscopic material design parameters. An efficient approximation is obtained by the reduced basis approach. A posteriori error estimates for the reduced forward problem are obtained in the periodic homogenization setting, using the so called two scale weak f...

We present a mathematical model for a study of the mechanical properties of endovascular stents in their expanded state. The model is based on the theory of slender curved rods. Stent struts are modeled as linearly elastic curved rods that satisfy the kinematic and dynamic contact conditions at the “vertices” where the struts meet. This defines a stent as a mesh of curved rods. A weak formulati...

∆u = ∂2u ∂x1 + ∂2u ∂x2 where x = (x1, x2) ∈ R2 are Cartesian coordinates. Even though the Poisson equation looks very special it is an important model case representing several problems from physics and engineering, e.g. electrostatics, stationary heat transfer and other diffusion problems. Variations of the techniques we will study apply to a wide class of second order so-called elliptic probl...