# Guaranteed error estimate for solutions to linear two-point boundary value problems with FEM

Akitoshi Takayasu, Shinichi Oishi, Takayuki Kubo

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

### Abstract

In this article, we consider a guaranteed error estimate procedure for solutions to linear two-point boundary value problems. 'Guaranteed' error estimate is rigorous, i.e. it takes into account every error such as the discretization error and the rounding error when solving the problems. It also enables us to prove the existence and the uniqueness of the exact solution. We define the solution operator to bound the guaranteed error. In order to get an approximate solution, the finite element method is used. The original problem is transformed into an operator equation. There are two main points in our proposal method. One is to compute the inverse operator estimation. This estimation is obtained by Theorem1. The other is to get the residual of the operator equation. We lead the estimation of the residual by a standard norm estimation. Finally, some numerical results are presented.

Original language English Asia Simulation Conference 2009, JSST 2009 Japan Society for Simulation Technology, JSST Published - 2009 Asia Simulation Conference 2009, JSST 2009 - ShigaDuration: 2009 Oct 7 → 2009 Oct 9

### Other

Other Asia Simulation Conference 2009, JSST 2009 Shiga 09/10/7 → 09/10/9

### Fingerprint

Two-point Boundary Value Problem
Boundary value problems
Error Estimates
Finite element method
Operator Equation
Mathematical operators
Discretization Error
Rounding error
Operator
Approximate Solution
Uniqueness
Exact Solution
Finite Element Method
Norm
Numerical Results

### Keywords

• Finite element method
• Guaranteed error estimate
• Two-point boundary value problem

### ASJC Scopus subject areas

• Modelling and Simulation

### Cite this

Takayasu, A., Oishi, S., & Kubo, T. (2009). Guaranteed error estimate for solutions to linear two-point boundary value problems with FEM. In Asia Simulation Conference 2009, JSST 2009 Japan Society for Simulation Technology, JSST.

Guaranteed error estimate for solutions to linear two-point boundary value problems with FEM. / Takayasu, Akitoshi; Oishi, Shinichi; Kubo, Takayuki.

Asia Simulation Conference 2009, JSST 2009. Japan Society for Simulation Technology, JSST, 2009.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Takayasu, A, Oishi, S & Kubo, T 2009, Guaranteed error estimate for solutions to linear two-point boundary value problems with FEM. in Asia Simulation Conference 2009, JSST 2009. Japan Society for Simulation Technology, JSST, Asia Simulation Conference 2009, JSST 2009, Shiga, 09/10/7.
Takayasu A, Oishi S, Kubo T. Guaranteed error estimate for solutions to linear two-point boundary value problems with FEM. In Asia Simulation Conference 2009, JSST 2009. Japan Society for Simulation Technology, JSST. 2009
Takayasu, Akitoshi ; Oishi, Shinichi ; Kubo, Takayuki. / Guaranteed error estimate for solutions to linear two-point boundary value problems with FEM. Asia Simulation Conference 2009, JSST 2009. Japan Society for Simulation Technology, JSST, 2009.
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