An Error Estimate for the Finite Element Solution of an Elliptic Problem with a Nonlinear Newton Boundary Condition in Nonpolygonal Domains

Abstract

The article is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition considered in a two-dimensional domain with a curved boundary. The existence and uniqueness of the weak solution of the continuous problem is a consequence of the monotone operator theory. The problem is discretized with the use of the finite element method. The main attention is paid to the effect of the approximation of the curved boundary by a piecewise linear boundary and of the evaluation of integrals by numerical quadratures. With the aid of some important properties of Zlámal's ideal triangulation and interpolation, the error estimate for the solution of the discrete problem is derived.

Keywords:

• Elliptic equation
• Nonlinear Newton boundary condition
• Monotone operator
• Finite element method
• Approximation of a curved boundary
• Numerical integration
• Ideal triangulation
• Ideal interpolation
• Convergence of the finite element method
• Error analysis
• AMS Subject Classifications: 65N30
• 65N15

Acknowledgment

Support of the Research Project J04/98/210000010 of Ministry of Education of the Czech Republic is gratefully acknowledged.