Iterative methods, preconditioning & numerical PDEs (IMET 2004). Preface
A multilevel method with correction by aggregation for solving discrete elliptic problems
The author studies the behaviour of a multi-level method that combines the Jacobi iterations and the correction by aggragation of unknowns. Our considerations are restricted to a simple one-dimensional example, which allows us to employ the technique of the Fourier analysis. Despite of this restriction we are able to demonstrate differences between the behaviour of the algorithm considered and of multigrid methods employing interpolation instead of aggregation.
Two simple derivations of universal bounds for the C.B.S. inequality constant
Universal bounds for the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for piecewise linear-linear and piecewise quadratic-linear finite element spaces in 2 space dimensions are derived. The bounds hold for arbitrary shaped triangles, or equivalently, arbitrary matrix coefficients for both the scalar diffusion problems and the elasticity theory equations.
Laudatio to Owe Axelsson
Efficient inexact Newton-like methods with application to problems of the deformation theory of plasticity
Newton-like methods are considered with inexact correction computed by some inner iterative method. Composite iterative methods of this type are applied to the solution of nonlinear systems arising from the solution of nonlinear elliptic boundary value problems. Two main quastions are studied in this paper: the convergence of the inexact Newton-like methods and the efficient control of accuracy in computation of the inexact correction. Numerical experiments show the efficiency of the suggested composite...
Algebraic preconditioning for Biot-Barenblatt poroelastic systems
Poroelastic systems describe fluid flow through porous medium coupled with deformation of the porous matrix. In this paper, the deformation is described by linear elasticity, the fluid flow is modelled as Darcy flow. The main focus is on the Biot-Barenblatt model with double porosity/double permeability flow, which distinguishes flow in two regions considered as continua. The main goal is in proposing block diagonal preconditionings to systems arising from the discretization of the Biot-Barenblatt...
Professor Ivo Marek (1933–2017)
Composite grid finite element method: Implementation and iterative solution with inexact subproblems
This paper concerns the composite grid finite element (FE) method for solving boundary value problems in the cases which require local grid refinement for enhancing the approximating properties of the corresponding FE space. A special interest is given to iterative methods based on natural decomposition of the space of unknowns and to the implementation of both the composite grid FEM and the iterative procedures for its solution. The implementation is important for gaining all benefits of the described...
Hierarchical FEM: Strengthened CBS inequalities, error estimates and iterative solvers
This paper describes natural decomposition of hierarchical finite element spaces, discusses a characterization of this decomposition via strengthened CBS inequality and uses this decomposition for development of hierarchical error estimates and iterative solution methods.
A comparison of deterministic and Bayesian inverse with application in micromechanics
The paper deals with formulation and numerical solution of problems of identification of material parameters for continuum mechanics problems in domains with heterogeneous microstructure. Due to a restricted number of measurements of quantities related to physical processes, we assume additional information about the microstructure geometry provided by CT scan or similar analysis. The inverse problems use output least squares cost functionals with values obtained from averages of state problem quantities...
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