The main aim of this paper is to apply the simplest anisotropic linear triangular finite element to solve the nonlinear Schrodinger equation (NLS). Firstly, the error estimate and superclose property with order O(h(2)...
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The main aim of this paper is to apply the simplest anisotropic linear triangular finite element to solve the nonlinear Schrodinger equation (NLS). Firstly, the error estimate and superclose property with order O(h(2)) about the Ritz projection are given based on an anisotropic interpolation property and high accuracy analysis of this element. Secondly, through establishing the relationship between the Ritz projection and interpolation, the superclose property of the interpolation is received. Thirdly, the global superconvergence with order O(h(2)) is derived by use of the interpolation post-processing technique. Finally, a numerical example is provided to verify the theoretical results. It is noteworthy that the main results obtained for anisotropic meshes herein cannot be deduced by only employing the interpolation or Ritz projection. (C) 2014 Elsevier Ltd. All rights reserved.
We explicitly determine the Babuska-Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The equation for determination is the transcendental equa...
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We explicitly determine the Babuska-Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The equation for determination is the transcendental equation t + tan t = 0, so that the solution call be numerically obtained with desired accuracy and verification. Such highly accurate approximate values for the constant can be widely used for a priori and a posteriori error estimations in adaptive computation and/or numerical verification.
In this paper, an effective numerical fully discrete finiteelement scheme for the distributed order time fractional diffusion equations is developed. By use of the composite trapezoid formula and the well-known L1 fo...
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In this paper, an effective numerical fully discrete finiteelement scheme for the distributed order time fractional diffusion equations is developed. By use of the composite trapezoid formula and the well-known L1 formula approximation to the distributed order derivative and linear triangular finite element approach for the spatial discretization, we construct a fully discrete finiteelement scheme. Based on the superclose estimate between the interpolation operator and the Ritz projection operator and the interpolation post-processing technique, the superclose approximation of the finiteelement numerical solution and the global superconvergence are proved rigorously, respectively. Finally, a numerical example is presented to support the theoretical results.
The goal of this paper is to discuss high accuracy analysis of a fully-discrete scheme for 2D multi-term time fractional wave equations with variable coefficient on anisotropic meshes by approximating in space by line...
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The goal of this paper is to discuss high accuracy analysis of a fully-discrete scheme for 2D multi-term time fractional wave equations with variable coefficient on anisotropic meshes by approximating in space by linear triangular finite element method and in time by Crank-Nicolson scheme. The stability is firstly proved unconditionally. In the analysis of superclose properties, how to deal with the item for variable coefficient is the main difficulty. In order to do this, a new projection operator is defined and the relationship between the proposed projection operator and interpolation operator about linear triangular finite element is deduced. Consequently, the global superconvergence result is obtained by use of interpolation postprocessing technique. The numerical examples show that the proposed numerical method is highly accurate and computationally efficient.
A fully discrete scheme is proposed for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation with variable coefficients on anisotropic meshes, where lineartriangularfinite elelment method (FEM)...
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A fully discrete scheme is proposed for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation with variable coefficients on anisotropic meshes, where lineartriangularfinite elelment method (FEM) is used for the spatial discretization and modified L1 approximation coupled with Crank-Nicolson scheme is applied to temporal direction. The mixed equation concerned contains a time-space coupled derivative which is very different from the previous literature. The stability is firstly obtained. Based on the property of the projection operator, the special relation between the projection operator and the interpolation operator of linear triangular finite element, the optimal error estimation and the superclose result are deduced. Then the global superconvergence property is derived by the interpolated postprocessing technique. Some numerical experiments are carried out to confirm the theoretical analysis on anisotropic meshes.
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