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A posteriori error estimation for arbitrary order FEM applied to singularly perturbed one-dimensional reaction-diffusion problems. (English) Zbl 1340.65163
Extending his earlier work [BIT 47, No. 2, 379–391 (2007; Zbl 1221.65175)] on the case $$r=1$$, the author establishes an a posteriori bound on the $$\infty$$-norm of the error in an $$r$$th order finite element method (FEM) approximation to the solution, $$u$$, of $$-\varepsilon ^2u''+cu=f$$ in $$(0,1)$$, satisfying $$u(0)=u(1)=0$$. Here $$c$$ and $$f$$ are piecewise continuous real-valued functions with $$c\geq \gamma ^2$$ where $$\varepsilon(\ll 1)$$ and $$\gamma$$ are positive constants.

##### MSC:
 65L70 Error bounds for numerical methods for ordinary differential equations 65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations 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 34B05 Linear boundary value problems for ordinary differential equations
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##### References:
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