Grid equidistribution for reaction-diffusion problems in one dimension. (English) Zbl 1089.65077
This paper deals with an adaptive numerical method for the linear reaction-diffusion two-point boundary value problem: \(-\varepsilon^2u''(x)+b(x)u(x)=f(x)\), \(x\in(0,1)\), \(u(0) = u(1) =0\), where \(b,f\in [0,1]\) and \(0<\beta <b(x)\leq \overline b\) on \([0,1]\).
Some facts about the solution \(u(x)\) and stability bounds for the computed solution \(u_N\) (on an arbitrary mesh) are listed. These bounds are used to provide strong heuristic evidence for the choice of monitor function to be equidistributed in an adaptive algorithm. Some numerical results are presented.
Some facts about the solution \(u(x)\) and stability bounds for the computed solution \(u_N\) (on an arbitrary mesh) are listed. These bounds are used to provide strong heuristic evidence for the choice of monitor function to be equidistributed in an adaptive algorithm. Some numerical results are presented.
Reviewer: Pavol Chocholatý (Bratislava)
MSC:
| 65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 65L70 | Error bounds for numerical methods for ordinary differential equations |
| 34B05 | Linear boundary value problems for ordinary differential equations |
| 34E15 | Singular perturbations for ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
Keywords:
reaction-diffusion problem; singular perturbation; adaptive mesh; monitor function; equidistribution; two-point boundary value problem; stability; numerical resultsReferences:
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