Finite element analysis of parabolic integro-differential equations of Kirchhoff type |
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Authors: | Lalit Kumar Sivaji Ganesh Sista Konijeti Sreenadh |
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Institution: | 1. Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India;2. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India |
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Abstract: | The aim of this paper is to study parabolic integro-differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results. |
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Keywords: | finite element method integro-differential equations linearized backward euler method linearized Crank–Nicolson method |
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