A Posteriori Error Analysis of the Crank-Nicolson Finite Element Method for Parabolic Integro-Differential Equations

Author(s)

Reddy, G. Murali Mohan ^[1] ; Sinha, Rajen Kumar ^[2] ; Cuminato, Jose Alberto ^[1]

Affiliation

^[1] Univ Sao Paulo Sao Carlos, Dept Appl Math & Stat, Inst Math & Comp Sci, POB 668, BR-13560970 Sao Carlos, SP - Brazil

^[2] Indian Inst Technol Guwahati, Dept Math, Gauhati 781039 - India

• Document type: Journal article
• Source: JOURNAL OF SCIENTIFIC COMPUTING; v. 79, n. 1, p. 414-441, APR 2019.
• Web of Science Citations: 0

Abstract

We study a posteriori error analysis for the space-time discretizations of linear parabolic integro-differential equation in a bounded convex polygonal or polyhedral domain. The piecewise linear finite element spaces are used for the space discretization, whereas the time discretization is based on the Crank-Nicolson method. The Ritz-Volterra reconstruction operator (IMA J Numer Anal 35:341-371, 2015), a generalization of elliptic reconstruction operator (SIAM J Numer Anal 41:1585-1594, 2003), is used in a crucial way to obtain optimal rate of convergence in space. Moreover, a quadratic (in time) space-time reconstruction operator is introduced to establish second order convergence in time. The proposed method uses nested finite element spaces and the standard energy technique to obtain optimal order error estimator in the L-infinity(L-2)-norm. Numerical experiments are performed to validate the optimality of the error estimators. (AU)

• FAPESP's process: 16/19648-9 - Efficient numerical solution of the inverse Stefan problems using the method of fundamental solutions
• Grantee: Gujji Murali Mohan Reddy
• Support Opportunities: Scholarships in Brazil - Post-Doctoral