We study existence and some properties of solutions of the nonlinear elliptic equation N(x,a(u))Lu = f in unbounded domains. The above method is not a variational problem. Our techniques involve fixed point arguments and Galerkin method.
We discuss a parallel implementation of the domain decomposition method based on the macro-hybrid formulation of a second order elliptic equation and on an approximation by the mortar element method. The discretization leads to an algebraic saddle- point problem. An iterative method with a block- diagonal preconditioner is used for solving the saddle- point problem. A parallel implementation of the method is emphasized. Finally the results of numerical experiments are presented.
We present and analyse in this paper a novel colocated Finite Volume scheme for the solution of the Stokes problem. It has been developed following two main ideas. On one hand, the discretization of the pressure gradient term is built as the discrete transposed of the velocity divergence term, the latter being evaluated using a natural finite volume approximation; this leads to a non-standard interpolation formula for the expression of the pressure on the edges of the control volumes. On the other...
A simple superconvergent scheme for the derivatives of finite element solution is presented, when linear triangular elements are employed to solve second order elliptic systems with boundary conditions of Newton’s or Neumann’s type. For bounded plane domains with smooth boundary the local -superconvergence of the derivatives in the -norm is proved. The paper is a direct continuations of [2], where an analogous problem with Dirichlet’s boundary conditions is treated.
Second order elliptic systems with boundary conditions of Dirichlet, Neumann’s or Newton’s type are solved by means of linear finite elements on regular uniform triangulations. Error estimates of the optimal order are proved for the averaged gradient on any fixed interior subdomain, provided the problem under consideration is regular in a certain sense.
Second order elliptic systems with Dirichlet boundary conditions are solved by means of affine finite elements on regular uniform triangulations. A simple averagign scheme is proposed, which implies a superconvergence of the gradient. For domains with enough smooth boundary, a global estimate is proved in the -norm. For a class of polygonal domains the global estimate can be proven.
We consider the local projection finite element method for the discretization of a scalar convection-diffusion equation with a divergence-free convection field. We introduce a new fluctuation operator which is defined using an orthogonal projection with respect to a weighted inner product. We prove that the bilinear form corresponding to the discrete problem satisfies an inf-sup condition with respect to the SUPG norm and derive an error estimate for the discrete solution.
We study a method based on Balancing Domain Decomposition by Constraints (BDDC) for numerical solution of a single-phase flow in heterogeneous porous media. The method solves for both flux and pressure variables. The fluxes are resolved in three steps: the coarse solve is followed by subdomain solves and last we look for a divergence-free flux correction and pressures using conjugate gradients with the BDDC preconditioner. Our main contribution is an application of the adaptive algorithm for selection...
We prove that penalization of constraints occuring in the linear elliptic Neumann problem yields directly the exact solution for an arbitrary set of penalty parameters. In this case there is a continuum of Lagrange's multipliers. The proposed penalty method is applied to calculate the magnetic field in the window of a transformer.