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

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

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

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

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

Robust Iterative Solvers for Gao Type Nonlinear Beam Models in Elasticity

The goal of this paper is to present various types of iterative solvers: gradient iteration, Newton’s method and a quasi-Newton method, for the finite element solution of elliptic problems arising in Gao type beam models (a geometrical type of nonlinearity, with respect to the Euler–Bernoulli hypothesis). Robust behaviour, i.e., convergence independently of the mesh parameters, is proved for these methods, and they are also tested with numerical experiments

Numerikus funkcionálanalízis

A funkcionálanalízis a matematikai analízisből kinőtt azon tudományág, melynek lényege végtelen dimenziós terek közti lineáris és nemlineáris leképezések vizsgálata. A benne megjelenő absztrakció lehetővé teszi az egységes tárgyalásmódot. E könyv témájának, a numerikus funkcionálanalízisnek a fogalma arra alapszik, hogy ezek az egységes, absztrakt módszerek éppoly alkalmasak a vizsgált egyenletek konstruktív megoldási algoritmusainak kidolgozására és analízisére, mint elméleti vizsgálatukra. E könyv megírásának mozgatórugója, hogy numerikus funkcionálanalízisről szóló könyv magyarul még nem elérhető. A könyv négy részből áll. Az I. részben a funkcionálanalízis egyes alapismereteit foglaljuk össze. A II. és III. rész lineáris, ill. nemlineáris operátoregyenletek megoldhatósági eredményeiről, azaz a megoldás fogalmáról, létezéséről és egyértelműségéről szól a szükséges elméleti háttérrel együtt. A IV. rész tartalmazza a különféle operátoregyenlet-típusokra vonatkozó közelítő módszerek tárgyalását. A vizsgált eljárások elsősorban két nagy csoportba tartoznak: projekciós, ill. iterációs módszerek. Ennek az anyagnak egy része megfelel az ELTÉ-n tartott funkcionálanalízis BSc és nemlineáris funkcionálanalízis MSc előadás témájának, az utolsó fejezet tárgya pedig újabb kutatásokhoz kapcsolódik

Harmonic averages, exact difference schemes and local Green's functions in variable coefficient PDE problems

A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in a Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes