Volume 55, pp. 652-670, 2022.

A time-stepping finite element method for a time-fractional partial differential equation of hidden-memory space-time variable order

Xiangcheng Zheng and Hong Wang

Abstract

We analyze a time-stepping finite element method for a time-fractional partial differential equation with hidden-memory space-time variable order. Due to the coupling of the space-dependent variable order with the finite element formulation and the hidden memory, the variable fractional order cannot be split from the space and destroys the monotonicity in the time-stepping discretization. Because of these difficulties, the numerical analysis of a fully-discrete finite element method of the proposed model remained untreated in the literature. We develop an alternative analysis to address these issues and to prove an optimal-order error estimate of the fully-discrete finite element scheme without any assumption on the regularity of the true solution and perform numerical experiments to substantiate the theoretical findings.

Full Text (PDF) [340 KB], BibTeX

Key words

fractional differential equation, well-posedness and regularity, hidden memory, space-time variable order, time-stepping finite element discretization, error estimate

AMS subject classifications

35R11, 65M12

Links to the cited ETNA articles

[10]	Vol. 5 (1997), pp. 1-6 Kai Diethelm: An algorithm for the numerical solution of differential equations of fractional order

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