Crank-Nicolson finite element methods for nonlocal problems with \(p\)-Laplace-type operator

Volume 33, Issue 1, pp 57--70 http://dx.doi.org/10.22436/jmcs.033.01.05

Publication Date: November 17, 2023 Submission Date: February 08, 2023 Revision Date: June 11, 2023 Accteptance Date: October 14, 2023

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Authors

M. S. D. Haggar - Department of Mathematics, University of Ndjamena, Chad. M. Mbehou - Department of Mathematics, University of Yaounde I, Cameroon. A. Njifenjou - Faculty of Industrial Engineering, University of Douala, Cameroon.

Abstract

A theoretical analysis of a Crank-Nicolson Galerkin finite element method for a class of nonlinear nonlocal diffusion problems associated with \(p\)-Laplace-type operator is presented here. It is shown, by a rigorous analysis that the unconditionally optimal error estimates for the fully discrete scheme are established. The presence of the nonlocal term in the models destroys the sparsity of the Jacobian matrices when solving the problem numerically using finite element method and Newton-Raphson method. As a consequence, computations consume more time and space in contrast to local problems. To overcome this difficulty, a new algorithm is proposed to avoid the full Jacobian matrix. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

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Keywords

• Crank-Nicolson scheme
• Newton-Raphson method
• Galerkin finite element method
• nonlocal diffusion term
• \(p\)-Laplace operator
• optimal error estimate

MSC

•  65N12
•  65N30
•  35K65
•  35J65

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