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**The h, p and h-p versions of the finite element method in 1 dimension. II. The error analysis of the h- and h-p versions.**
*(English)*
Zbl 0614.65089

This paper concentrates on h and h-p versions of FEM. Generally, the h- version of FEM has the degree of elements fixed and the convergence is achieved by the refinement of the mesh. The h-p version combines both approaches, the p-version and h-version.

The same model problem as in Part I [ibid. 49, 577-612 (1986; reviewed above)] is considered. The paper is mainly concerned with the relation between the relative error in the energy norm and the number of degrees of freedom. The authors show that the selection of the mesh and degree of elements is essential for the performance of the method. More exactly, the proper selection of the h-p version leads to the exponential rate of convergence while the h-version with improper mesh, gives very low algebraic rate when a singularity is present.

The same model problem as in Part I [ibid. 49, 577-612 (1986; reviewed above)] is considered. The paper is mainly concerned with the relation between the relative error in the energy norm and the number of degrees of freedom. The authors show that the selection of the mesh and degree of elements is essential for the performance of the method. More exactly, the proper selection of the h-p version leads to the exponential rate of convergence while the h-version with improper mesh, gives very low algebraic rate when a singularity is present.

Reviewer: C.-I.Gheorghiu

### MSC:

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

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

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

34B05 | Linear boundary value problems for ordinary differential equations |

### Keywords:

mesh refinement; finite element method; h-p versions; h-version; convergence; relative error; exponential rate of convergence; singularity### Citations:

Zbl 0614.65088
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\textit{W. Gui} and \textit{I. Babuška}, Numer. Math. 49, 613--657 (1986; Zbl 0614.65089)

### References:

[1] | Babuška, I., Gui, W.: Theh, p andh-p Versions of the Finite Element Method for One Dimensional Problem. Part I: The Error Analysis of thep-Version. Numer. Math.49, 577–612 (1986) · Zbl 0614.65088 |

[2] | DeVore, R., Scherer, K.: Variable knot variable degree spline approximation tox {\(\beta\)}. In: Quantitative Approximation. Proceedings, Bonn, pp. 101–131 (1979) |

[3] | Scherer, K.: On optimal global error bounds obtained by scaled local error estimates. Numer. Math.36, 257–277 (1981) · Zbl 0495.65006 |

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