Variational Problems in Domains with Cusp-Points and the Finite Element Method

ABSTRACT

A variational problem in a two-dimensional domain with cusp-points corresponding to a linear elliptic boundary value problem is formulated and the unique existence of its solution is proved. The corresponding finite element method using triangular finite C ⁰-elements with polynomials of the first degree is analyzed and both the convergence (under the assumptions sufficient for the existence of the exact solution) and the maximal rate of convergence 𝒪(h) are proved.

Keywords:

• Convergence
• Error estimates
• Existence and uniqueness theorem
• Finite element method
• Variational problems in bounded two-dimensional domains with cusp-points

MSC Classification 2000:

• 65N30
• 65N15
• 65N99

ACKNOWLEDGMENTS

This work was supported by grants GACR 201/03/0570 of the Grant Agency of the Czech Republic and MSM 262100001 of the Ministry of Education of the Czech Republic.