Convergence of Three-Step Taylor Galerkin Finite Element Scheme Based Monotone Schwarz Iterative Method for Singularly Perturbed Differential-Difference Equation

Abstract

Monotone Schwarz iterative methods for parabolic partial differential equations are well known for their advantage of eliminating the search for an initial solution. In this article, we propose a monotone Schwarz iterative method for singularly perturbed parabolic retarded differential-difference equations based on a three-step Taylor Galerkin finite element scheme. The stability and ε-uniform convergence of the three-step Taylor Galerkin finite element method have been discussed. Further, by using maximum principle and induction hypothesis, the convergence of the proposed monotone Schwarz iterative method has been established.

Keywords:

• Boundary layer
• Exponentially fitted splines
• Finite element method
• Mass-lumped
• Maximum principle
• Monotone Schwarz iterative method (MSIM)
• Singularly perturbed problems (SPP)
• Taylor Galerkin method

Mathematics Subject Classification:

• 35K20
• 65L11
• 65M12