Abstract P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary re- sults for 1D case, condition number estimates and some inequalities for 2D reference element.

Cell migration plays an important role in numerous normal and pathological processes. The physical mechanisms of adhesion, protrusion/extension, contractions, and polarization can regulate cell migration speed, persistence time, and downstream effects in paracrine and endocrine signaling. Methods for understanding these biophysical and biochemical responses to date have been limited to the use of external forces acting on mechanotransductive receptors. Additionally, as the use of magnetic nanoparticles for cell tracking and cell manipulation studies continues to gain popularity, so does the importance of understanding the cellular response to mechanical forces caused by these magnetic systems. This thesis work utilizes superparamagnetic iron oxide nanoparticles and static magnets to induce an endogenous magnetic force on the cell membrane. This cell manipulation model is used to better understand the mechanobiological responses of mesenchymal stem cell to SPIO labeling and endogenous force generation. Directionally persistent motility, cytoskeletal reorganization, and altered pro-migratory cytokine secretion is reported in this thesis as a response to SPIO based cell manipulation.

Anisotropic local interpolation error estimates are derived for quadrilateral and hexahedral Lagrangian finite elements with straight edges. These elements are allowed to have diameters with different asymptotic behaviour in different space directions. The case of affine elements (parallelepipeds) with arbitrarily high degree of the shape functions is considered first. Then, a careful examination of the multi-linear map leads to estimates for certain classes of more general, isoparametric elements. As an application, the Galerkin finite element method for a reaction diffusion problem in a polygonal domain is considered. The boundary layers are resolved using anisotropic trapezoidal elements.

For two-point boundary value problems in weak formulation with piecewise constant coefficients and piecewise continuous right-hand side functions we derive a representation of its weak solution by local Green's functions. Then we use it to generate exact three-point discretizations by Galerkin's method on essentially arbitrary grids. The coarsest possible grid is the set of points at which the piecewise constant coefficients and the right- hand side functions are discontinuous. This grid can be refined to resolve any solution properties like boundary and interior layers much more correctly. The proper basis functions for the Galerkin method are entirely defined by the local Green's functions. The exact discretizations are of completely exponentially fitted type and stable. The system matrices of the resulting tridiagonal systems of linear equations are in any case irreducible M-matrices with a uniformly bounded norm of its inverse.

For solving systems of grid equations approximating elliptic boundary value problems a method of constructing variable preconditioning procedures is presented. The main purpose is to discuss how an efficient preconditioning iterative procedure can be constructed in the case of elliptic problems with disproportional coefficients, e.g. equations with a large coefficient in the reaction term (or a small diffusion coefficient). The optimality of the suggested technique is based on fictitious space and multilevel decom- position methods. Using an additive form of the preconditioners, we intro- duce factors into the preconditioners to optimize the corresponding conver- gence rate. The optimization with respect to these factors is used at each step of the iterative process. The application of this technique to two-level $p$-hierarchical precondi- tioners and domain decomposition methods is considered too.

A new backward stable, structure preserving method of complexity O(n^3) is presented for computing the stable invariant subspace of a real Hamiltonian matrix and the stabilizing solution of the continuous-time algebraic Riccati equation. The new method is based on the relationship between the invariant subspaces of the Hamiltonian matrix H and the extended matrix /0 H\ and makes use \H 0/ of the symplectic URV-like decomposition that was recently introduced by the authors.

Rank-revealing ULV and URV factorizations are useful tools to determine the rank and to compute bases for null-spaces of a matrix. However, in the practical ULV (resp. URV ) factorization each left (resp. right) null vector is recomputed from its corresponding right (resp. left) null vector via triangular solves. Triangular solves are required at initial factorization, refinement and updating. As a result, algorithms based on these factorizations may be expensive, especially on parallel computers where triangular solves are expensive. In this paper we propose an alternative approach. Our new rank-revealing ULV factorization, which we call ¨top-down¨ ULV factorization ( TDULV -factorization) is based on right null vectors of lower triangular matrices and therefore no triangular solves are required. Right null vectors are easy to estimate accurately using condition estimators such as incremental condition estimator (ICE). The TDULV factorization is shown to be equivalent to the URV factorization with the advantage of circumventing triangular solves.

This paper is concerned with a specific finite element strategy for solving elliptic boundary value problems in domains with corners and edges. First, the anisotropic singular behaviour of the solution is described. Then the finite element method with anisotropic, graded meshes and piecewise linear shape functions is investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates for functions from anisotropically weighted spaces are derived. Finally, a numerical experiment is described, that shows a good agreement of the calculated approximation orders with the theoretically predicted ones.

This paper deals with parallelization methods for time-dependent problems where the time steps are shared out among the processors. A Full Multigrid technique serves as solution algorithm, hence information of the preceding time step and of the coarser grid is necessary to compute the solution at each new grid level. Applying the usual extrapolation formula to process this information, the parallelization will not be very efficient. We developed another extrapolation technique which causes a much higher parallelization effect. Test examples show that no essential loss of exactness appears, such that the method presented here shall be well-applicable.

The paper gives an overview over local inequalities for anisotropic simplicial Lagrangian finite elements. The main original contributions are the estimates for higher derivatives of the interpolation error, the formulation of the assumptions on admissible anisotropic finite elements in terms of geometrical conditions in the three-dimensional case, and an anisotropic variant of the inverse inequality. An application of anisotropic meshes in the context of a stabilized Galerkin method for a convection-diffusion problem is given.