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**Self-adaptive strategy for one-dimensional finite element method based on element energy projection method.**
*(English)*
Zbl 1231.65121

Summary: Based on the newly-developed element energy projection (EEP) method for computation of super-convergent results in one-dimensional finite element method (FEM), the task of self-adaptive FEM analysis was converted into the task of adaptive piecewise polynomial interpolation. As a result, a satisfactory FEM mesh can be obtained, and further FEM analysis on this mesh would immediately produce an FEM solution which usually satisfies the user specified error tolerance. Even though the error tolerance was not completely satisfied, one or two steps of further local refinements would be sufficient. This strategy was found to be very simple, rapid, cheap and efficient. Taking the elliptical ordinary differential equation of second order as the model problem, the fundamental idea, implementation strategy and detailed algorithm are described. Representative numerical examples are given to show the effectiveness and reliability of the proposed approach.

### 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:

finite element method (FEM); self-adaptive solution; super-convergence; element energy projection; ordinary differential equation (ODE)### Software:

COLSYS
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\textit{S. Yuan} and \textit{X.-F. He}, Appl. Math. Mech., Engl. Ed. 27, No. 11, 1461--1474 (2006; Zbl 1231.65121)

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### References:

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