The backward euler anisotropic a posteriori error analysis for parabolic integro‐differential equations. Issue 5 (21st January 2016)

• Record Type: Journal Article
• Authors: Reddy, Gujji Murali Mohan; Sinha, Rajen Kumar
• Is Part Of: Numerical methods for partial differential equations. Volume 32:Issue 5(2016:Sep.)
• Page Start: 1309
• Page End: 1330
• Publication Date: 2016-01-21
• Subjects: parabolic integro‐differential equations -- finite element method -- fully discrete scheme -- optimal -- anisotropic error estimator; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22049
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract : We derive two optimal a posteriori error estimators for an implicit fully discrete approximation to the solutions of linear integro‐differential equations of the parabolic type. A continuous, piecewise linear finite element space is used for the space discretization and the time discretization is based on an implicit backward Euler method. The a posteriori error indicator corresponding to space discretization is derived using the anisotropic interpolation estimates in conjunction with a Zienkiewicz‐Zhu error estimator to approach the error gradient. The error due to time discretization is derived using continuous, piecewise linear polynomial in time. We use the linear approximation of the Volterra integral term to estimate the quadrature error in the second estimator. Numerical experiments are performed on the isotropic mesh to validate the derived results.© 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1309–1330, 2016