A maximum principle for some nonlinear cooperative elliptic PDE systems with mixed boundary conditions

One of the classical maximum principles state that any nonnegative solution of a proper elliptic PDE attains its maximum on the boundary of a bounded domain. We suitably extend this principle to nonlinear cooperative elliptic systems with diagonally dominant coupling and with mixed boundary conditions. One of the consequences is a preservation of nonpositivity, i.e. if the coordinate functions or their uxes are nonpositive on the Dirichlet or Neumann boundaries, respectively, then they are all nonpositive on the whole domain as well. Such a result essentially expresses that the studied PDE system is a qualitatively reliable model of the underlying real phenomena, such as proper reaction-diffusion systems in chemistry

Quasi-Newton variable preconditioning for nonlinear nonuniformly monotone elliptic problems posed in Banach spaces

Quasi-Newton-type iterative solvers are developed for a wide class of nonlinear elliptic problems. The presented generalization of earlier efficient methods covers various nonuniformly elliptic problems arising, e.g., in non-Newtonian flows or for certain glaciology models. The robust estimates are reinforced by several examples

Preconditioning of block tridiagonal matrices

Preconditioning methods via approximate block factorization for block tridiagonal matrices are studied. Bounds for the resulting condition numbers are given, and two methods for the recursive construction of the approximate Schur complements are presented. Illustrations for elliptic problems are also given, including a study of sensitivity to jumps in the coefficients and of a suitably motidied Poincaré-Steklov operator on the continuous level

Sobolev gradient preconditioning for the electrostatic potential equation

AbstractSobolev gradient type preconditioning is proposed for the numerical solution of the electrostatic potential equation. A constructive representation of the gradients leads to efficient Laplacian preconditioners in the iteration thanks to the available fast Poisson solvers. Convergence is then verified for the corresponding sequence in Sobolev space, implying mesh independent convergence results for the discretized problems. A particular study is devoted to the case of a ball: due to the radial symmetry of this domain, a direct realization without discretization is feasible. The simplicity of the algorithm and the fast linear convergence are finally illustrated in a numerical test example

Some discrete maximum principles arising for nonlinear elliptic finite element problems

The discrete maximum principle (DMP) is an important measure of the qualitative reliability of the applied numerical scheme for elliptic problems. This paper starts with formulating simple sufficient conditions for the matrix case and for nonlinear forms in Banach spaces. Then a DMP is derived for finite element solutions for certain nonlinear partial differential equations: we address nonlinear elliptic problems with mixed boundary conditions and interface conditions, allowing possibly degenerate nonlinearities and thus extending our previous results

Discretization error estimates in maximum norm for convergent splittings of matrices with a monotone preconditioning part

For finite difference matrices that are monotone, a discretization error estimate in maximum norm follows from the truncation errors of the discretization. It enables also discretization error estimates for derivatives of the solution. These results are extended to convergent operator splittings of the difference matrix where the major, preconditioning part is monotone but the whole operator is not necessarily monotone

Reaching the superlinear convergence phase of the CG method

The rate of convergence of the conjugate gradient method takes place in essen- tially three phases, with respectively a sublinear, a linear and a superlinear rate. The paper examines when the superlinear phase is reached. To do this, two methods are used. One is based on the K-condition number, thereby separating the eigenval- ues in three sets: small and large outliers and intermediate eigenvalues. The other is based on annihilating polynomials for the eigenvalues and, assuming various an- alytical distributions of them, thereby using certain refined estimates. The results are illustrated for some typical distributions of eigenvalues and with some numerical tests