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Computational decomposition and composition technique for approximate solution of nonstationary problems

P. N. Vabishchevich

TL;DR

A general approach is proposed to construct decomposition-composition algorithms for evolution equations in finite-dimensional Hilbert spaces based on two main variants of the decomposition of the unit operator in the corresponding spaces at the decomposition stage and the application of additive operator-difference schemes at the composition stage.

Abstract

Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional equations can be reduced by additive decomposition of the problem operator(s) and composition of the approximate solution using particular explicit-implicit time approximations. Such a technique is currently applied in conditions where the decomposition step is uncomplicated. A general approach is proposed to construct decomposition-composition algorithms for evolution equations in finite-dimensional Hilbert spaces. It is based on two main variants of the decomposition of the unit operator in the corresponding spaces at the decomposition stage and the application of additive operator-difference schemes at the composition stage. The general results are illustrated on the boundary value problem for a second-order parabolic equation by constructing standard splitting schemes on spatial variables and region-additive schemes (domain decomposition schemes).

Computational decomposition and composition technique for approximate solution of nonstationary problems

TL;DR

A general approach is proposed to construct decomposition-composition algorithms for evolution equations in finite-dimensional Hilbert spaces based on two main variants of the decomposition of the unit operator in the corresponding spaces at the decomposition stage and the application of additive operator-difference schemes at the composition stage.

Abstract

Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional equations can be reduced by additive decomposition of the problem operator(s) and composition of the approximate solution using particular explicit-implicit time approximations. Such a technique is currently applied in conditions where the decomposition step is uncomplicated. A general approach is proposed to construct decomposition-composition algorithms for evolution equations in finite-dimensional Hilbert spaces. It is based on two main variants of the decomposition of the unit operator in the corresponding spaces at the decomposition stage and the application of additive operator-difference schemes at the composition stage. The general results are illustrated on the boundary value problem for a second-order parabolic equation by constructing standard splitting schemes on spatial variables and region-additive schemes (domain decomposition schemes).
Paper Structure (23 sections, 5 theorems, 121 equations, 3 figures)

This paper contains 23 sections, 5 theorems, 121 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Cauchy problem
  4. Decomposition-composition technique
  5. Model problem
  6. Differential Problem
  7. Approximation over space
  8. Decomposition
  9. Composition
  10. Decomposition in one space
  11. Additive decomposition of the unit operator
  12. The first variant of decomposition
  13. Second decomposition variant
  14. Other spaces
  15. Decompositions for a set of spaces
  16. ...and 8 more sections

Key Result

Theorem 1

The scheme with weight (2.5), (2.6) is unconditionally stable at $\sigma \geq 1/2$ in $H_D$, $D = A, I, A^{-1}$. For the solution, the estimate is holds.

Figures (3)

  • Figure 1: Grid: $\bullet$ --- internal nodes, $\circ$ --- boundary nodes
  • Figure 2: Calculated nodes for $A_1$ (left) and for $A_2$ (right).
  • Figure 3: Grid when the region is partitioned into rectangles (left) and when skeletonized (right): $\omega_1$ --- blue nodes, $\omega_2$ --- red nodes.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5