and Norms Error Estimates in Finite Element Method for Linear Parabolic Interface Problems

Abstract

We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L ²(Ω) and H ¹(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose.

Keywords:

• Finite element method
• Interface
• Optimal
• Parabolic
• Pointwise-in-time error estimates
• Semidiscrete and fully discrete schemes

AMS Subject Classification:

• 65N30
• 65F10

ACKNOWLEDGMENTS

The author is grateful to the anonymous referee for his valuable comments and suggestions which greatly improved the presentation of this article.