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  • 教师姓名： 王飞
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  • 所在单位： 数学与统计学院
  • 学历： 硕博连读
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  • 性别： 男
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  • 学位： 博士
  • 职称： 教授
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Discontinuous Galerkin Methods for Variational Inequalities

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Many problems in physical and engineering sciences are modeled by partial differential equations (PDEs). However, various more complex physical processes are described by variational inequalities (VIs). Variational inequalities form an important and very useful class of nonlinear problems arising in diverse application areas of physical, engineering, financial, and management sciences.a

 

Discontinuous Galerkin (DG) Methods for VIs

 
Discontinuous Galerkin (DG) methods have been widely used for solving a variety of mathematical and physical problems due to the advantage of capturing non-smooth or oscillatory solutions effectively and the flexibility in constructing feasible local shape function spaces. Because of their discontinuous nature, DG methods are very suitable for hp-adaptive algorithms. Another advantage is the locality of the discretization, which makes them ideally suited for parallel computing. Moreover, DG methods incorporate non-homogeneous boundary conditions in the weak formulations, which greatly increases the robustness and accuracy of boundary condition implementation. In the past five years, Dr. Wang have worked with Professor Weimin Han (University of Iowa), Professor Xiaoliang Cheng (Zhejiang University) and other collaborators on developing DG methods for solving variational inequalities. Due to inequality form of the problems, the Galerkin orthogonality is lost when DG methods are applied to solve VIs, resulting in substantial difficulty in analyzing DG methods for VIs. Also, the bilinear forms of DG schemes have coercivity only in the finite element spaces, not in the original function space. Therefore, the standard analytical techniques of finite element methods for VIs are not applicable for DG cases. We had to search for a new approach. We studied numerous DG methods for solving elliptic variational inequalities of both the first and second kinds, and established a unified a priori error analysis. We proved that the consistent and stable DG schemes can achieve the optimal convergence order with linear element for various variational inequality problems, such as obstacle problem ([1]) , frictional contact problem ([1]), Signorini problem ([2]), and quasi-static contact problem ([3]). 
 

[1] Fei Wang, Weimin Han and Xiaoliang Cheng, Discontinuous Galerkin Methods for Solving Elliptic Variational Inequalities, SIAM J Numerical Analysis, 48 (2010), 708-733.      

[2] Fei Wang, Weimin Han and Xiaoliang Cheng, Discontinuous Galerkin Methods for Solving Signorini Problem, IMA Journal of Numerical Analysis, 31 (2011), 1754-1772.      

[3] Fei Wang, Weimin Han and Xiaoliang Cheng, Discontinuous Galerkin Methods for Solving the Quasistatic Contact Problem, Numerische Mathematik, 126 (2014), 771-800.

 

Adaptive Finite Element Methods for VIs

 

Compared with uniform mesh, adaptive finite element method can improve precision with smaller memory and less time cost. DG methods are very suitable for hp-adaptive algorithm. Thus, it is a natural choice to study adaptive DG methods for VIs. For the DG schemes of obstacle problems, we derived reliable error estimators of the residual type. The efficiency of the estimators is theoretically explored and numerically confirmed ([4]). It is difficult to develop a posteriori error estimates to variational inequalities due to the inequality feature. The idea was to use Lagrange multiplier to transfer obstacle problem to the associated elliptic PDE, then apply a posteriori error theory of elliptic differential equation to derive reliable error estimators. This idea was extended to give a posteriori error analysis for VIs of the second kind. Moreover, a proof was provided for the efficiency of the error estimators, which was first proved in ([5]). A posteriori error estimates of DG methods for a frictional contact problem was studied, and we proved that the error estimators are reliable and efficient. The following two figures are for one numerical example about adaptive DG for solving obstacle problem. Figure 1 shows the adaptive meshes with hanging nodes, N denoting the number of elements. In Figure 2, the L2 norm and H1 norm errors are given respectively on L2 norm error are 0.097020 and 0.001293, respectively. To achieve the same level of accuracy, with adaptivity, we only need to compute the solution on the mesh $T11$; correspondingly N = 3975, and H1 norm and L2 norm errors being 0.088184 and 0.001134, respectively. Therefore comparing to the uniform refinement, with smaller memory, the adaptive strategy saves lots of computing time. 

 

Figure 1. Adaptive Refined Meshes 

                                        Figure 1. Adaptive Refined Meshes                                                                            Figure 2. Errors on Adaptive and Uniform Meshes

[4] Fei Wang, Weimin Han, Joseph Eichholz and Xiaoliang Cheng, A Posteriori Error Estimates of Discontinuous Galerkin Methods for Obstacle Problems, Nonlinear Analysis: Real World Applications, 22 (2015), 664-679.       

[5] Fei Wang and Weimin Han, Another View for a Posteriori Error Estimates for Variational Inequalities of the Second Kind, Applied Numerical Mathematics, 72 (2013), 225-233.

 

C0 DG Methods for 4th-order Variational Inequalities

 

To solve 4th order elliptic PDEs, conforming finite element (FE) method uses C1 finite elements, which are not easy to construct and implement. To resolve this problem, nonconforming FE method have been developed. However, the nonconforming FE space needs to be carefully chosen so that the inconsistent error can be controlled. Instead, we can consider C0 DG methods to solve 4th order PDEs. The FE spaces belong to C0, instead of C1, and at the meantime, penalty terms for derivative between element boundary are added to force the derivative almost continuous. Therefore, C0 DG schemes have good accuracy with less number of freedom, so a lot of time  can be saved in solving the discretized problems. We study C0 DG methods to solve elliptic variational inequalities of 4th-order for the Kirchhoff plates. It is difficult to construct stable DG methods for such problems because of the higher order and the inequality form. For the frictional contact Kirchhoff plates, we proved that the quadratic C0 DG schemes achieve the optimal convergence order ([6]).

 

[6] Fei Wang, Tianyi Zhang and Weimin Han, C0 Discontinuous Galerkin Methods for a Plate Contact Problem, Journal of Computational Mathematics, 37 (2019), 184-200.

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