A posteriori Error Analysis of Finite Element methods for Parabolic Integro-differential Equations

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The aim of the thesis is to study a posteriori error analysis of finite element method for linear parabolic integro-differential equations (PIDEs) in a convex polygonal or polyhedral domain. PIDEs and their variants arise in various applications, such as heat conduction in material with memory, the compression of poro-viscoelasticity media, nuclear reactor dynamics and the epidemic phenomena in biology. Since PIDE may be thought of as a perturbation of the purely parabolic problem, it is therefore, natural to expect how the a posteriori error analysis of parabolic problems can be extended to PIDEs. Such an extension is not straightforward in the presence of Volterra integral term. In isotropic settings, we derive a posteriori error estimators for both the spatially semidis- crete and the fully discrete (backward Euler and Crank-Nicolson) schemes for PIDEs. A novel space-time reconstruction operator is introduced which is an a posteriori counterpart of the Ritz-Volterra projection. Moreover, this reconstruction operator is a generalization of the elliptic reconstruction operator and we call it as Ritz-Volterra reconstruction operator. The Ritz-Volterra reconstruction operator is used in a crucial way to derive optimal order a pos- teriori error estimates in the L1(L2)-norm. The related a posteriori error estimates for the Ritz-Volterra reconstruction error are also established. For the Crank-Nicolson scheme, the derivation of the a posteriori error estimator relies essentially on the Ritz-Volterra reconstruc- tion operator and a novel space-time quadratic (in time) reconstruction operator. To reduce the number of degrees of freedom and computational effort to achieve the same convergence as compared to the isotropic meshes, we also consider the a posteriori error anal- ysis for PIDE in an anisotropic framework. We derive a posteriori error estimators for both fully discrete backward Euler and Crank-Nicolson schemes for PIDEs in the L2(H1)-norm in a two dimensional convex polygonal domain. The a posteriori error indicators corresponding to space discretizations are derived using the anisotropic interpolation estimates in conjunc- tion with a Zienkiewicz-Zhu error estimator to approach the error gradient. The error due to time discretization is derived using continuous, piece wise linear polynomial in time in case of backward Euler scheme, whereas to recover optimality for Crank-Nicolson scheme we intro- duce continuous, piecewise quadratic time reconstructions, namely, Crank-Nicolson memory reconstruction and three point reconstruction.
Supervisor: Rajen Kumar Sinha