Some new methods for numerical solution of initial value problems
Algebraic approach to domain decomposition
An iterative procedure containing two parameters for solving linear algebraic systems originating from the domain decomposition technique is proposed. The optimization of the parameters is investigated. A numerical example is given as an illustration.
Об одном принципе сходимости в пространстве Гильберта
On a construction of fast direct solvers
Fast direct solvers for the Poisson equation with homogeneous Dirichlet and Neumann boundary conditions on special triangles and tetrahedra are constructed. The domain given is extended by symmetrization or skew symmetrization onto a rectangle or a rectangular parallelepiped and a fast direct solver is used there. All extendable domains are found. Eigenproblems are also considered.
Eigenvalues and eigenfunctions of the Laplace operator on an equilateral triangle for the discrete case
A discretized boundary value problem for the Laplace equation with the Dirichlet and Neumann boundary conditions on an equilateral triangle with a triangular mesh is transformed into a problem of the same type on a rectangle. Explicit formulae for all eigenvalues and all eigenfunctions are given.
Eigenvalues and eigenfunctions of the Laplace operator on an equilateral triangle
A boundary value problem for the Laplace equation with Dirichlet and Neumann boundary conditions on an equilateral triangle is transformed to a problem of the same type on a rectangle. This enables us to use, e.g., the cyclic reduction method for computing the numerical solution of the problem. By the same transformation, explicit formulae for all eigenvalues and all eigenfunctions of the corresponding operator are obtained.
Universally optimal approximation of functionals
A universal optimal in order approximation of a general functional in the space of continuous periodic functions is constructed and its fundamental properties and some generalizations are investigated. As an application the approximation of singular integrals is considered and illustrated by numerical results.
A two parameter iterative method for solving algebraic systems of domain decomposition type
An iterative procedure containing two parameters for linear algebraic systems originating from the domain decomposition technique is proposed. The optimization of the parameters is investigated. A numeric example is given as an illustration.
An iterative method of alternating type for systems with special block matrices
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.
The method of fictitious right-hand sides
The paper deals with the application of a fast algorithm for the solution of finite-difference systems for boundary-value problems on a standard domain (e.g. on a rectangle) to the solution of a boundary-value problem on a domain of general shape contained in the standard domain. A simple iterative procedure is suggested for the determination of fictitious right-hand sides for the system on the standard domain so that its solution is the desired one. Under the assumptions that are usual for matrices...
The general form of local bilinear functions
The scalar product of the FEM basis functions with non-intersecting supports vanishes. This property is generalized and the concept of local bilinear functional in a Hilbert space is introduced. The general form of such functionals in the spaces and is given.
Sur une modification de la méthode de M. Kantorovitch
Zprávy. Ivo Babuška laureátem státní ceny 1968 v oboru matematiky
Schwarzův algoritmus pro polyharmonické funkce
Jak počítače počítají
Spolehlivost numerických výpočtů
Overimplicit methods for the solution of evolution problems
Overimplicit multistep methods
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