Uniformly superconvergent analysis of an efficient two-grid method for nonlinear Bi-wave singular perturbation problem

Shi, Dongyang; Wu, Yanmi

doi:10.1016/j.amc.2019.124772

Cited by 8 publications

(4 citation statements)

References 21 publications

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“…Now we will turn to analyze the well‐posedness of the approximation scheme (5) through the Brouwer fixed point theorem [6, 18, 19]. Theorem For any constant 0 < δ ≤ 1, problem (5) has a unique solution ψ h .…”

Section: Well‐posedness Of the Approximation Schemementioning

confidence: 99%

“…Now we will turn to analyze the well-posedness of the approximation scheme (5) through the Brouwer fixed point theorem [6,18,19].…”

Section: Well-posedness Of the Approximation Schemementioning

confidence: 99%

“…For example, two conforming Galerkin finite element methods (FEMs) and the modified nonconforming Morley-type Galerkin FEMs were proposed for the linear bi-wave equations, and convergence error estimates were derived in [2,3], respectively. The Ciarlet-Raviart mixed FEMs were analyzed in [4][5][6] for the linear bi-wave equations and problem (1), and the uniform superconvergence estimates in the broken H 1 norm were deduced, respectively. We mention that the piecewise polynomials space was required to be C 1 continuous in [2], which would be expensive and less efficient.…”

Section: Introductionmentioning

confidence: 99%

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Quasi‐uniform convergence analysis of a modified penalty finite element method for nonlinear singularly perturbed bi‐wave problem

Shi

2020

Numerical Methods Partial

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In this paper, a modified penalty finite element method (FEM) for solving the nonlinear singularly perturbed bi‐wave problem with a rectangular Morley element is presented. The second order term, the nonlinear term, and the source term are approximated by the bilinear interpolation part (conforming part) instead of the whole function on each element, respectively. The well‐posedness of the approximation solution is proved through Brouwer fixed point theorem. Quasi‐uniform optimal error estimate in the energy norm is derived independent of the negative powers of the real perturbation parameter δ appearing in the considered problem, which improves the corresponding result in the existing literature. Finally, some numerical results are given to verify the theoretical analysis.

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Section: Well‐posedness Of the Approximation Schemementioning

confidence: 99%

“…Now we will turn to analyze the well-posedness of the approximation scheme (5) through the Brouwer fixed point theorem [6,18,19].…”

Section: Well-posedness Of the Approximation Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quasi‐uniform convergence analysis of a modified penalty finite element method for nonlinear singularly perturbed bi‐wave problem

Shi

2020

Numerical Methods Partial

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“…Hence, 0 < δ < 1 is expected to be small for d-wave superconductors and problem (1.1) degenerates into the semilinear parabolic equation when δ → 0. In recent years, there are some theoretical analysis and numerical simulations about FEMs, such as optimal order error estimates of conforming Galerkin FEMs and the modified Morleytype discontinuous Galerkin FEMs in [10,11], uniform superconvergence error estimates of Ciarlet-Raviart schemes with the conforming and nonconforming elements in [12][13][14]. But these work mainly focused on the stationary singularly perturbed Bi-wave problems.…”

Section: Introductionmentioning

confidence: 99%

Uniform Superconvergence Analysis of a Two-Grid Mixed Finite Element Method for the Time-Dependent Bi-Wave Problem Modeling $D$-Wave Superconductors

Shi¹

2024

JCM

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In this paper, a two-grid mixed finite element method (MFEM) of implicit Backward Euler (BE) formula is presented for the fourth order time-dependent singularly perturbed Bi-wave problem for d-wave superconductors by the nonconforming EQ rot 1 element. In this approach, the original nonlinear system is solved on the coarse mesh through the Newton iteration method, and then the linear system is computed on the fine mesh with Taylor's expansion. Based on the high accuracy results of the chosen element, the uniform superclose and superconvergent estimates in the broken H 1 -norm are derived, which are independent of the negative powers of the perturbation parameter appeared in the considered problem. Numerical results illustrate that the computing cost of the proposed two-grid method is much less than that of the conventional Galerkin MFEM without loss of accuracy.

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Quasi-uniform and unconditional superconvergence analysis of Ciarlet–Raviart scheme for the fourth order singularly perturbed Bi-wave problem modelingd-wave superconductors

Shi

2021

Applied Mathematics and Computation

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Uniformly superconvergent analysis of an efficient two-grid method for nonlinear Bi-wave singular perturbation problem

Cited by 8 publications

References 21 publications

Quasi‐uniform convergence analysis of a modified penalty finite element method for nonlinear singularly perturbed bi‐wave problem

Quasi‐uniform convergence analysis of a modified penalty finite element method for nonlinear singularly perturbed bi‐wave problem

Uniform Superconvergence Analysis of a Two-Grid Mixed Finite Element Method for the Time-Dependent Bi-Wave Problem Modeling $D$-Wave Superconductors

Quasi-uniform and unconditional superconvergence analysis of Ciarlet–Raviart scheme for the fourth order singularly perturbed Bi-wave problem modelingd-wave superconductors

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