Abstract. In this article we are interested in the derivation of efficient domain decomposition methods for the viscous primitive equations of the ocean. We consider the rotating 3d incompressible hydrostatic Navier-Stokes equations with free surface. Performing an asymptotic analysis of the system in the regime of small Rossby numbers, we compute an approximate Dirichlet to Neumann operator and build an optimized Schwarz waveform relaxation algorithm. We established that the algorithm is well defined and provide numerical evidences of the convergence of the method.
SUMMARYIn order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we present here a method which consists in inverting, on a ÿnite element space, a non-singular integral operator. This operator is a geometrical perturbation of the Steklov operator, and we precisely deÿne the relation between the geometrical perturbation and the dimension of the ÿnite element space, in order to obtain a stable and convergent scheme. Furthermore, this numerical scheme does not give rise to any singular integral.The scheme can also be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single layer potential, the special quadrature formula being deÿned by the introduction of a neighbouring curve.In the present paper, we prove stability and we give error estimates of our numerical scheme when the Laplace problem is set on a disk. We will extend our results to any domains by using compact perturbation arguments, in a second paper.
We prove that under some global conditions on the maximum and the minimum eigenvalue of the matrix of the coefficients, the gradient of the (weak) solution of some degenerate elliptic equations has higher integrability than expected. Technically we adapt the Giaquinta-Modica regularity method in some degenerate cases. When the dimension is two, a consequence of our result is a new Hölder continuity result for the weak solution.
We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solution are considered and compared. We establish existence and regularity results. In particular we study the boundedness of weak solutions. We also establish an existence result for a classical solution.
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