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عنوان انگلیسی مقاله:
A Petrov–Galerkin finite element method for variable-coefficient fractional diffusion equations
سال انتشار : 2015
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مقدمه انگلیسی مقاله:
1. Introduction
In the last few decades fractional differential equations (FDEs) have found increasingly more applications in fluid mechanics [1], anomalous diffusion and acceleration of steep fronts in reaction–diffusion processes [2,3], turbulence in geophysical flows or plasma physics [4–6], continuum mechanics [7], as they provide very effective alternatives for modeling complex systems characterized by nonlocal phenomena and long range interactions. However, FDEs present mathematical difficulties that have not been encountered in the context of second-order differential equations. In their pioneer work [8], Ervin and Roop proved coercivity of a Galerkin formulation and the well-posedness of the homogeneous Dirichlet boundary-value problem of a constant–coefficient conservative FDE. We showed that for variable-coefficient FDEs the Galerkin formulation loses its coercivity [9] and that the Galerkin finite element methods might fail to converge [10].To overcome these difficulties we proposed a Petrov–Galerkin formulation for the homogeneous Dirichlet boundary-value problem of FDEs, and proved its weak coercivity and well-posedness [9]. However, there is a sharp difference between a Galerkin formulation and a Petrov–Galerkin formulation: Coercivity of a Galerkin formulation on an infinite-dimensional admissible space ensures that of the formulation on any finite-dimensional subspace. Consequently, the unique solvability and stability of Galerkin finite element methods are guaranteed automatically. In contrast, weak coercivity of a Petrov–Galerkin formulation on a pair of infinite-dimensional product spaces cannot ensure that of the formulation on any pair of finite-dimensional subspaces. Therefore, one still has to analyze how to choose appropriate finite-dimensional trial space and test space to ensure the weak coercivity and so the unique solvability and stability of the corresponding Petrov–Galerkin finite element method. In this paper we utilize the DPG (discontinuous Petrov–Galerkin) framework of Demkowicz and Gopalakrishnan [11–14] to develop a Petrov–Galerkin finite element method for a class of variable-coefficient conservative FDEs in one space dimension. We prove its error estimate in the energy norm and the L 2 norm. Numerical experiments are presented to verify the convergence rates of the method. The rest of the paper is organized as follows: In Section 2 we present the model problem and cite known results to be used subsequently. In Section 3 we apply the DPG framework to the model problem. In Section 4 we develop a Petrov–Galerkin finite element method with optimal test functions for fractional diffusion equations with a constant diffusivity coefficient. We then prove the corresponding error estimates. In Section 5 we develop a Petrov–Galerkin finite element method with approximately optimal test functions for fractional diffusion equations with a variable diffusivity coefficient and prove the corresponding error estimates in the energy norm and the L 2 norm. In Section 6 we conduct numerical experiments to investigate the performance of the Petrov–Galerkin method and to verify its convergence rate numerically. In Section 7 we draw concluding remarks and outline future work.
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کلمات کلیدی:
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