Unconditional optimal error estimates of a two-grid method for semilinear parabolic equation

2018

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In this paper, the stabilized mixed finite element methods are presented for the Navier‐Stokes equations with damping. The existence and uniqueness of the weak solutions are proven by use of the Brouwer fixed‐point theorem. Then, optimal error estimates for the H1‐norm and L2‐norm of the velocity and the L2‐norm of the pressure are derived. Moreover, on the basis of the optimal L2‐norm error estimate of the velocity, a stabilized two‐step method is proposed, which is more efficient than the usual stabilized methods. Finally, two numerical examples are implemented to confirm the theoretical analysis.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

2018

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“…Similar to Shi and Yang, we set τ 2 = h 2 , H 4 = h 2 and use Newton iterations with 1 e −10 as the tolerance error to solve nonlinear system . Tables 1 to 3 show that the errors of

false‖ I_{h} u^{n} - Ũ_{h}^{n} ‖_{1}

and

false‖ u^{n} - I_{2 h}^{2} Ũ_{h}^{n} ‖_{1}

at different time level are convergent with order O ( h 2 ).…”

Section: Numerical Experimentsmentioning

confidence: 99%

Superconvergence of a two‐grid method for fourth‐order dispersive‐dissipative wave equations

Wang

2019

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In this paper, the superconvergence analysis of a two‐grid method (TGM) with low‐order finite elements is presented for the fourth‐order dispersive‐dissipative wave equations for a second order fully discrete scheme. The superclose estimates in the H1‐norm on the two grids are obtained by the combination technique of the interpolation and Ritz projection. Then, with the help of the interpolated postprocessing technique, the global superconvergence properties are deduced. Finally, numerical results are provided to show the performance of the proposed TGM for conforming bilinear element and nonconforming EQ1rot element, respectively. It shows that the TGM is an effective method to the problem considered of our paper compared with the traditional Galerkin finite element method (FEM).

“…Now, we estimate the terms on the right side of (17 Here, we give the following mathematical induction hypothesis to estimate B 4 and B 5 , and this technique can be found in previous studies 13,14,31,33,34 :…”

Section: Optimal Error Estimate For the Linearized Fully Discrete Schemementioning

confidence: 99%

Nonconforming quadrilateral finite element method for nonlinear Kirchhoff‐type equation with damping

2019

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Nonconforming quadrilateral EQ1rot finite element method (FEM) of nonlinear Kirchhoff‐type equation with damping is studied on anisotropic meshes. Based on the property of the nonlocal term of this equation, unconditional optimal error estimates of Ofalse(hfalse) and Ofalse(h+τ2false) ( h, the spatial parameter, and τ, the time step) in the broken H1 norm are deduced for the semidiscrete and a linearized fully discrete schemes without any restrictions of τ through a distinct approach compared with the methods used for other partial differential equations, respectively. Besides, the damping term appearing in the Kirchhoff‐type equation is solved with a novel technique, which is the major difficulty in the theoretical analysis. Finally, some numerical results are provided to verify the theoretical analysis.