Weak Galerkin finite element method for the parabolic integro-differential equation with weakly singular kernel

Zhou, Jun; Xu, Dan; Dai, Xiuxiu

doi:10.1007/s40314-019-0807-7

Cited by 8 publications

(4 citation statements)

References 21 publications

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“…After equations (8) and (9) have been linearized, according to equation (12), they can be rewritten as…”

Section: Optimization Mathematical Model For Node_face Frictional Conmentioning

confidence: 99%

“…According to Figure 2, when the penalty factor α varies within the range between 10 and 10 8 , the objective function value will be close to zero, namely, the nonpenetrating constraints expressed in equation (7) can be satisfied and the system will be stable. So, the penalty factor α can be taken as 10 8 . While penalty factor α is larger than 10 8 , the objective function value will increase sharply, namely, equation 7cannot be satisfied, and the "spring" will carry out punishments on the contact.…”

Section: Selection Of Penalty Factor From Equations (3) and (23)mentioning

confidence: 99%

“…For the contact problems, only very few of them can be solved by analytical methods, and most of them need to be simulated by numerical methods such as the Finite Element Method (FEM) [3,4] and the Boundary Element Method (BEM) [5,6]. e FEM is relatively mature and widely used [7][8][9][10]. However, the BEM has the advantages of dimension reduction, singularity adaptation, high precision, and so on [11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

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A Kind of FM-BEM Penalty Function Method for a 3D Elastic Frictional Contact Nonlinear System

Jing

et al. 2021

Journal of Mathematics

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In this paper, a kind of node_face frictional contact FM-BEM penalty function method is presented for 3D elastic frictional contact nonlinear problems. According to the principle of minimum potential energy, nonpenetrating constraints are introduced into the elastic frictional contact system as a penalty term. By using the least square method and penalty function method, an optimization mathematical model and a mathematical programming model with a penalty factor are established for the node_face frictional contact nonlinear system. For the two models, a penalty optimization IGMRES (m) algorithm is proposed, and the influences of different penalty factors on the solution of the whole system are analyzed. Finally, a numerical simulation is carried out for two elastic frictional contact objects, and some important results including displacements, pressures, friction forces, and friction slips in the contact area are presented. Theoretical analysis and numerical experiment show that the newly presented FM-BEM penalty function method not only is efficient and practical but also has much superiority. It is easy to implement, and it is fast convergent with good stability.

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“…After equations (8) and (9) have been linearized, according to equation (12), they can be rewritten as…”

Section: Optimization Mathematical Model For Node_face Frictional Conmentioning

confidence: 99%

Section: Selection Of Penalty Factor From Equations (3) and (23)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Kind of FM-BEM Penalty Function Method for a 3D Elastic Frictional Contact Nonlinear System

Jing

et al. 2021

Journal of Mathematics

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“…The weak Galerkin (WG) finite element method is a numerical technique for solving partial differential equations. Since it has been proposed by Wang [28], the WG method has been applied successfully to the discretization of several classes of partial differential equations and variational inequalities, e.g., second-order elliptic problems [2-5, 8, 9, 11, 16, 19, 29, 31], the Stokes equations [25,30,33,34], the Biharmonic equations [18,21,27,35], the Maxwell equations [22], the Oseen equations [15], the Helmholtz equations [20,23], the multi-term time-fractional diffusion equations [36], the parabolic integro-differential equations [37], the interface problems [17,24], the natural convection problems [6], and the elliptic variational inequality [26].…”

Section: Introductionmentioning

confidence: 99%

A Robust Modified Weak Galerkin Finite Element Method for Reaction-Diffusion Equations

Li¹,

null²,

Huang³

2022

NMTMA

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In this paper, a robust modified weak Galerkin (MWG) finite element method for reaction-diffusion equations is proposed and investigated. An advantage of this method is that it can deal with the singularly perturbed reaction-diffusion equations. Another advantage of this method is that it produces fewer degrees of freedom than the traditional WG method by eliminating the element boundaries freedom. It is worth pointing out that, in our method, the test functions space is the same as the finite element space, which is helpful for the error analysis. Optimalorder error estimates are established for the corresponding numerical approximation in various norms. Some numerical results are reported to confirm the theory.

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A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model

Qiu

Guo

et al. 2019

Numer Algor

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Weak Galerkin finite element method for the parabolic integro-differential equation with weakly singular kernel

Cited by 8 publications

References 21 publications

A Kind of FM-BEM Penalty Function Method for a 3D Elastic Frictional Contact Nonlinear System

A Kind of FM-BEM Penalty Function Method for a 3D Elastic Frictional Contact Nonlinear System

A Robust Modified Weak Galerkin Finite Element Method for Reaction-Diffusion Equations

A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model

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