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On the robust exponential convergence of $$hp$$ finite element methods for problems with boundary layers. (English) Zbl 0887.65106
The author concentrates on approximation of the Dirichlet problem for the singularly perturbed ordinary differential equation $$-d^2u''+b(x)u=f(x),$$ on the interval $$[-1,1]$$, where $$b$$ and $$f$$ are analytic functions, $$b$$ is bounded away from zero and $$d>0$$ is a constant. The author is interested in methods which exhibit exponential convergence in the case that the degree $$p$$ of the approximating elements is not large compared with $$1/d$$. The results in this paper extend the work of C. Schwab and M. Suri [Math. Comput. 65, No. 216, 1403-1429 (1996; Zbl 0853.65115)] to include analytical functions $$f$$.
Since the problem is analytic, exponential convergence is assumed for large $$p$$. For smaller $$p$$, some accomodation must be made for boundary layers near the ends of the intervals. Schwab and Suri employed a mesh consisting of three elements, two small elements at the ends of the interval and the remaining middle. Such meshes are employed in this paper also, along with meshes consisting of several elements of geometrically decreasing size as they get nearer the endpoints. Exponential convergence is proved for these geometrically decreasing meshes as well as for the three element meshes.
Several numerical examples are presented, graphically illustrating the theoretical convergence rates.

##### MSC:
 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 35B25 Singular perturbations in context of PDEs 35J25 Boundary value problems for second-order elliptic equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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