A recent multiplicity result by Ricceri, stated for equations in Hilbert spaces, is extended to a wider class of Banach spaces. Applications to nonlinear boundary value problems involving the p-Laplacian are presented.
We prove the existence of a maximum principle for operators of the type , for weights with subharmonic. It is associated with certain simply connected subdomains that are produced by a Hele-Shaw flow emanating from a given point in the domain. For constant weight, these are the circular disks in the domain. The principle is equivalent to the following statement. THEOREM. Suppose is logarithmically subharmonic on the unit disk, and that the weight times area measure is a reproducing measure...
An iterative procedure for systems with matrices originalting from the domain decomposition technique is proposed. The procedure introduces one iteration parameter. The convergence and optimization of the method with respect to the parameter is investigated. The method is intended not as a preconditioner for the CG method but for the independent use.
We examine an elliptic equation in a domain Ω whose boundary ∂Ω is countably (m-1)-rectifiable. We also assume that ∂Ω satisfies a geometrical condition. We are interested in an overdetermined boundary value problem (examined by Serrin [Arch. Ration. Mech. Anal. 43 (1971)] for classical solutions on domains with smooth boundary). We show that existence of a solution of this problem implies that Ω is an m-dimensional Euclidean ball.
We show that a non-standard mixed finite element method proposed by Barrios and Gatica in 2007, is a higher order perturbation of the least-squares mixed finite element method. Therefore, it is also superconvergent whenever the least-squares mixed finite element method is superconvergent. Superconvergence of the latter was earlier investigated by Brandts, Chen and Yang between 2004 and 2006. Since the new method leads to a non-symmetric system matrix, its application seems however more expensive...
Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite and is...
So far optimal error estimates on Bakhvalov-type meshes are only known for finite difference and finite element methods solving linear convection-diffusion problems in the one-dimensional case. We prove (almost) optimal error estimates for problems with exponential boundary layers in two dimensions.
Patch substructuring methods are non-overlapping domain decomposition methods like classical substructuring methods, but they use information from geometric patches reaching into neighboring subdomains condensated, on the interfaces to enhance the performance of the method, while keeping it non-overlapping. These methods are very convenient to use in practice, but their convergence properties have not been studied yet. We analyze geometric patch substructuring methods for the special case of one...