A numerical method for computing singular minimizers

A numerical method called element removal method is designed to calculate singular minimizers which cannot be approximated by simple applications of standard numerical methods because of the so-called Lavrentiev phenomenon. The convergence of the method is proved. The results of numerical experiments show that the method is effective.

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ELEMENT REMOVAL METHOD FOR SINGULAR MINIMIZERS IN PROBLEMS OF HYPERELASTICITY

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202595000231 ◽

1995 ◽

Vol 05 (03) ◽

pp. 387-399 ◽

Cited By ~ 6

Author(s):

ZHIPING LI

Keyword(s):

Finite Element ◽

Numerical Method ◽

Lavrentiev Phenomenon ◽

Removal Method ◽

Finite Element Approximations ◽

Element Removal Method ◽

Element Removal ◽

Admissible Functions ◽

Singular Minimizers

A numerical method called element removal method is applied to calculate singular minimizers in problems of hyperelasticity. The method overcomes the difficulty in finite element approximations caused by restrictions, such as det (I+∇u)>0, on admissible functions and can avoid Lavrentiev phenomenon if it does occur in the problem. The convergence of the method is proved.

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A TRUNCATION METHOD FOR DETECTING SINGULAR MINIMIZERS INVOLVING THE LAVRENTIEV PHENOMENON

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202506001376 ◽

2006 ◽

Vol 16 (06) ◽

pp. 847-867 ◽

Cited By ~ 4

Author(s):

YU BAI ◽

ZHI-PING LI

Keyword(s):

Numerical Method ◽

Variational Problems ◽

Numerical Results ◽

Two Dimensional ◽

Dimensional Problem ◽

Minimizing Sequences ◽

Lavrentiev Phenomenon ◽

Absolute Minimizers ◽

Sobolev Exponent ◽

Singular Minimizers

A numerical method using the truncation technique on the integrand is developed for computing singular minimizers or singular minimizing sequences in variational problems involving the Lavrentiev phenomenon. It is proved that the method can detect absolute minimizers with various singularities whether the Lavrentiev phenomenon is involved or not. It is also proved that, when the absolute infimum is not attainable, the method can produce minimizing sequences. Numerical results on Manià's example and a two-dimensional problem involving the Lavrentiev phenomenon with continuous Sobolev exponent dependence, are given to show the efficiency of the method.

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