Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction-diffusion problem.

*(English)*Zbl 1221.65175This paper is concerned with the numerical solution of singularly perturbed one-dimensional boundary value problems of the type
\[
- \varepsilon^2 u'' + r u = f \text{ in } (0,1), u(0)= u(1)=0,
\]
where \( r,f \in {\mathcal C}^2[0,1]\), \( 0 << \varepsilon < 1\), \( r(x) > \rho^2 > 0 \) for \( x \in [0,1]\), whose solution possess two layers of width \({\mathcal O} ( \varepsilon \log \varepsilon^{-1} )\) at the endpoints of the domain. The author proposes an FE discretization in an arbitrary mesh \( ( x_j )_{j=0}^N \) in \([0,1]\), where the test function \( \varphi_i\) is the standard hat function associated to the \(i\)-th mesh node. It is found that this discretization leads to a finite difference scheme that – unlike the standard central difference schemes – is not inverse monotone. In this context the author proves that it is stable in the maximum norm, and for the layer adapted meshes proposed by Shishkin and Bakhvalov some convergence results are established. The paper ends with the numerical results obtained for two test problems with the above meshes.

Reviewer: Manuel Calvo (Zaragoza)

##### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

##### Keywords:

two-point boundary value problems; singular perturbations; stability of non-monotone FEM discretization in maximum norm; convergence in layer adapted meshes##### References:

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