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**Toward a universal h-p adaptive finite element strategy. I: Constrained approximation and data structure.**
*(English)*
Zbl 0723.73074

The present article is the first one of the trilogy of papers [see the following entries (Zbl 0723.73075; Zbl 0723.73076)] on the development of an adaptive h-p version (h: sizes of the elements; p: orders of the shape functions) of the finite element method for the solution of various boundary value problems in solid and fluid mechanics. For the h-p method represents significant departures from conventional finite element techniques, a resolution of several formidable problems in their effective implementation is required, such as new data structures, equation solvers and certain criteria for choosing a distribution of mesh sizes and approximate orders.

In Part 1 of the presentation, general formulation issues are taken up, the data structure is developed, and the h-p adaptive strategy is introduced, thus contributing to a finite element scheme for linear elliptic boundary-value problems characterized by general elliptic systems of partial differential equations. In particular, the h- and p- adaptive is discussed, restricting the irregularity of the considered meshes to the index one. Furthermore, the concept of constrained approximation is outlined and its impact on such basic ingredients to the FEM as element stiffness matrix and load vector calculations is presented. The following discussion on the h-p adaptive finite element method is then restricted to the two-dimensional case, where the initial mesh is topologically a portion of a regular, rectangular grid in \({\mathbb{R}}^ 2\). Finally, some details concerning the data structure are given. The paper concludes with a presentation of four illustrative examples and some details on forms of the corresponding variational formulations.

In Part 1 of the presentation, general formulation issues are taken up, the data structure is developed, and the h-p adaptive strategy is introduced, thus contributing to a finite element scheme for linear elliptic boundary-value problems characterized by general elliptic systems of partial differential equations. In particular, the h- and p- adaptive is discussed, restricting the irregularity of the considered meshes to the index one. Furthermore, the concept of constrained approximation is outlined and its impact on such basic ingredients to the FEM as element stiffness matrix and load vector calculations is presented. The following discussion on the h-p adaptive finite element method is then restricted to the two-dimensional case, where the initial mesh is topologically a portion of a regular, rectangular grid in \({\mathbb{R}}^ 2\). Finally, some details concerning the data structure are given. The paper concludes with a presentation of four illustrative examples and some details on forms of the corresponding variational formulations.

Reviewer: W.Ehlers (Essen)

### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

76M10 | Finite element methods applied to problems in fluid mechanics |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74P10 | Optimization of other properties in solid mechanics |

### Keywords:

data structures; linear elliptic boundary-value problems; general elliptic systems of partial differential equations; concept of constrained approximation; two-dimensional case
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\textit{L. Demkowicz} et al., Comput. Methods Appl. Mech. Eng. 77, No. 1--2, 79--112 (1989; Zbl 0723.73074)

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### References:

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[7] | L. Demkowicz and J.T. Oden, A review of local mesh refinement techniques and corresponding data structures in h-type adaptive finite element methods, TICOM Rept. 88-02, The Texas Institute for Computational Mechanics, The University of Texas at Austin, Texas 78712. |

[8] | Bank, R.E.; Sherman, A.H.; Weiser, A., Refinement algorithms and data structures for regular mesh refinement, (), 3-17 |

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.