An hp-version error analysis of the discontinuous Galerkin method for linear elasticity
Introduction
The linear elastic equations are used to describe the deformation of elastic structures under the action of prescribed loads, which are the fundamental equations in mathematical physics. Both the displacement and stress fields are the fundamental physical quantities in mechanical analysis. When approximating the displacement and stress simultaneously, the mixed finite element method (cf. [1], [2], [4], [12], [23], [24], [26], [27]) can achieve higher-accuracy stress than the standard displacement finite element method, with which the stress is obtained by differentiating the displacement and using the constitutive law of stress-strain. The main and critical difficulty in construction of such mixed methods is largely due to the fact that the stress tensor is symmetric and belong to . To overcome his difficulty, the discontinuous Galerkin (DG) finite element method is an apt choice to solve the linear elasticity by weakening the regularity of finite element space. Historically, some discontinuous Galerkin methods for linear elasticity are presented in primal formulation, including the local DG (LDG) method in [29], the compact DG method in [28], and the interior penalty DG method in [20], [21]. A mixed discontinuous Galerkin finite element method with symmetric stress tensor for linear elasticity is given in [10], which is a special case of general DG formulation in [13]. More recently, some stabilized mixed finite element methods with symmetric stress tensor for linear elasticity, based on the Hu–Zhang element in [23], [25], [26], are introduced in [12]. In [9], a mixed DG formulation is also designed but the stress tensor is nonsymmetric. Following the ideas in [11], [16], a general framework of constructing DG methods with symmetric stress tensor has been developed in [13] for solving the linear elasticity problem, and the h-optimal convergence of the resulting LDG method is developed as well.
Polynomials of arbitrary degree can be taken on each element in the discontinuous Galerkin method, for the continuity of finite element spaces across the interfaces of triangulation is not required. Thus it is natural to analyze DG method in hp-version context. To this end, we first review the hp-version finite element method for second order elliptic problems. The optimal convergence of the hp-version finite element method with the triangulation containing triangles and parallel elements in two dimension is developed in [7] for second order elliptic problems with smooth solutions. We refer to [34] for an excellent historical survey. Later on, based on the framework of the Jacobi-weighted Besov spaces (cf. [5], [6]), the optimal convergence of the hp-version finite element method with triangulation containing curvilinear triangles and quadrilaterals in two dimension is established in [18], [19] for problems with both smooth and singular solutions. On the other hand, hp-version error estimates are also developed for the DG method of second order problems in recent years (cf. [22], [31]), which are optimal in the mesh size h and suboptimal in the degree of polynomial p. In [17], under the condition that the exact solution of the reaction-diffusion equation belongs to an augmented Sobolev space, hp-optimal error estimates have been deduced for interior penalty DG method with triangulation containing elements being -diffeomorphic to parallelograms. And in [35], by virtue of continuous interpolations of the exact solution, a class of hp-version DG methods on parallelograms' mesh for Poisson's equation with homogeneous Dirichlet boundary condition have been proved to converge optimally in the energy norm with respect to both the local element sizes and polynomial degrees.
However, to the best of our knowledge, there are few results about hp-version mixed finite element methods for linear elasticity. Based on the Hellinger–Reissner formulation, a hp-mixed finite element space with symmetric stress tensor in two dimensions is constructed in [4], which is the first stable one using polynomial shape functions. After establishing the elasticity complex starting from the de Rham complex, Arnold–Falk–Winther element method for the modified Hellinger–Reissner formulation in three space dimensions is devised in [3], whose stress tensor is nonsymmetric. Furthermore, by designing projection based interpolation operators, Arnold–Falk–Winther element for linear elasticity is extended to meshes with elements of variable order in [32], [33]. We mention in passing that all the error analyses in these literatures just involve the h-version error analysis.
In this paper, we intend to develop the hp-version error analysis for the general mixed DG method for the linear elastic problem. To this end, we first derive the hp-version error estimates of two projection operators. Then incorporated with the techniques in [11] technically, we are able to obtain the hp-version error estimates for the previous method in energy norm and norm, respectively. Finally, a variety of numerical examples are provided for validating the theoretical results.
The rest of this paper is organized as follows. Some notations and the DG method in mixed formulation for linear elasticity are presented in Section 2. The hp-version error analysis for the DG method is given in Section 3. And in Section 4, a series of numerical results are included to show the numerical performance of the DG method proposed.
Section snippets
The DG method for linear elasticity
Assume that is a bounded polygon or polyhedron. Let be the stress, the displacement and the applied force. Denote by the linearized strain tensor with , tr the trace operator, and div the divergence operator. Consider linear elasticity in the stress-displacement formulation: where is the compliance tensor of fourth order defined by Here,
The hp-version error analysis for the DG method
In this section, we are going to establish hp-version error estimates for the DG method (3)–(4). Our derivation is mainly based on the techniques developed in [11]. To this end, we first rewrite (3)–(4) in a compact form, described as follows.
Find such that for all , where In the following, we always assume that is the solution of the original problem (1). Let be projection
Numerical results
In this section, we intend to present a variety of numerical examples in order to illustrate the numerical performance of the mixed DG method (7) (or equivalently, the method (3)–(4)). In all the numerical examples, we choose and . For any , we take and where . Set when , and let .
Conclusion
The hp-version error analysis is systematically developed for the mixed DG method (7) (or equivalently, the method (3)–(4)). The derivation is mainly based on the ideas in [11] and the hp-version error estimates of two projection operators. According to our numerical experiments, we may achieve the following conclusions:
- (1)
If , the error estimates of and in Theorem 1, Theorem 2 are sharp except the case .
- (2)
If , the error estimate of
Acknowledgements
The authors would like to express their sincere thanks to three anonymous reviewers whose comments and suggestions greatly improved an early version of the paper. The work of the first author was partly supported by the National Natural Science Foundation of China (Grant Nos. 11571237 and 11171219). The work of the second author was partly supported by the National Natural Science Foundation of China (Grant Nos. 11771338 and 11301396), Zhejiang Provincial Natural Science Foundation of China
References (35)
- et al.
Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche's method
Comput. Methods Appl. Mech. Eng.
(2002) - et al.
On the stability of the boundary trace of the polynomial -projection on triangles and tetrahedra
Comput. Math. Appl.
(2014) - et al.
Mixed hp-finite element method for linear elasticity with weakly imposed symmetry
Comput. Methods Appl. Mech. Eng.
(2009) - et al.
A mixed finite element method for elasticity in three dimensions
J. Sci. Comput.
(2005) - et al.
A family of higher order mixed finite element methods for plane elasticity
Numer. Math.
(1984) - et al.
Mixed finite element methods for linear elasticity with weakly imposed symmetry
Math. Comput.
(2007) - et al.
Mixed finite elements for elasticity
Numer. Math.
(2002) - et al.
Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces. I. Approximability of functions in the weighted Besov spaces
SIAM J. Numer. Anal.
(2001/02) - et al.
Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces. II. Optimal rate of convergence of the p-version finite element solutions
Math. Models Methods Appl. Sci.
(2002) - et al.
The version of the finite element method with quasi-uniform meshes
RAIRO Modél. Math. Anal. Numér.
(1987)