Research Article | Open Access
A Reduced Local Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity
A reduced local discontinuous Galerkin (RLDG) method for nearly incompressible linear elasticity is proposed in this paper, which is locking-free. RLDG method can be formally regarded as a special case of LDG method with . However, RLDG method is actually not covered by LDG method, where must be chosen to be positive to ensure the stability of LDG method. RLDG method can also be considered as the localization of some symmetric nonconforming mixed finite element method. The implementation of RLDG method is discussed. By introducing a lifting operator as LDG method, RLDG method can be rewritten as primal formulation with unknown displacement only. Next, we obtain that the convergence rates of the approximation to stress tensor in energy norm and displacement in -norm are and , respectively, which are both uniform with respect to . Moreover, we obtain a -conforming displacement by projecting the displacement and corresponding numerical trace of RLDG method into the Raviart-Thomas element space. And then we analyze the error estimates of this postprocessed displacement in -seminorm and -norm, which are also uniform with respect to . Finally, some numerical results are shown to demonstrate the theoretical results.
Volume locking phenomenon will happen when solving nearly incompressible linear elasticity problem by standard displacement method using continuous lower order finite elements (cf. [1–3]); that is, the convergence rates for these lower order conforming finite element methods will deteriorate as the Lamé constant tends to . For this reason, robust numerical methods with respect to are necessary to deal with volume locking phenomenon, and these methods are also called locking-free methods. As for standard displacement method, Babuška and Suri showed detailedly in  that it is locking-free whenever the polynomial order of the local shape space is no less than and locked for on triangular meshes, and locking phenomenon always takes place for any on quadrilateral meshes. Besides higher order -version conforming finite element methods, -version finite element method in , nonconforming finite element methods in [5–8], and mixed finite element methods in [9–11] are also suggested to overcome the volume locking phenomenon.
On the other hand, discontinuous Galerkin (DG) methods which possess the flexibility of constructing feasible local shape function spaces and the advantage of capturing non-smooth or oscillatory solutions effectively have been applied to solve nearly incompressible linear elasticity. Hansbo and Larson analyzed symmetric interior penalty discontinuous Galerkin (SIPDG) methods in [12, 13], whereas a dimensionless penalty parameter appeared in the stabilized term cannot be quantified in theory and can only be determined by numerical experience. A locking-free nonsymmetric interior penalty discontinuous Galerkin (NIPDG) method was developed by Wihler in . And Wihler also designed an adaptive algorithm for NIPDG method in , whose robust a posteriori error analysis was given by Bridgeman and Wihler in [15, 16]. A local discontinuous Galerkin (LDG) method based on a first-order system with three mathematical equations was presented and discussed by Cockburn et al. in , and a post-processed (div)-conforming displacement was achieved by projecting the displacement and corresponding trace of LDG method into the Raviart-Thomas element space. Using symmetric normal-normal continuous element for stress and tangential continuous Nédélec element for displacement, Pechstein and Schberl proposed a locking-free mixed finite element method in , which is in fact some kind of discontinuous Galerkin method. Through introducing a DG derivative, a family of discontinuous Galerkin mixed methods were designed by Shen and Lew in  which were proved to be locking-free as well. They also observed that locking might appear if Dirichlet boundary conditions were imposed strongly rather than weakly.
In this paper, we intend to propose a reduced local discontinuous Galerkin (RLDG) method for nearly incompressible linear elasticity. RLDG method can be formally regarded as a special case of LDG method proposed in  with . However, RLDG method is actually not covered by LDG method, where must be chosen to be positive to ensure the stability of LDG method, whereas RLDG method drops this requirement by choosing large enough finite element space for stress to match finite element space for displacement. On the other hand, RLDG method can also be considered as the localization of the symmetric nonconforming mixed finite element method given in . Based on this observation, the unique solvability of RLDG method is proved using the similar argument as the method in . Since there is not any continuity across the interior faces for the finite element space for stress, Schur complement is easy to obtain, and thus implementation of RLDG method is not difficult. On the other hand, RLDG method can be rewritten as primal formulation with unknown displacement only through introducing a lifting operator as LDG method in . According to the error equations, an interpolation operator designed in  and BDM interpolation operator, we obtain that the convergence rate of the approximation to stress tensor and displacement in energy norm is . And by duality argument, the convergence rate of the approximation to displacement in -norm is shown to be . These two convergence rates are both uniform with respect to ; hence our RLDG method is locking-free. Moreover, we obtain a (div)-conforming displacement by projecting the displacement and corresponding numerical trace of RLDG method into the Raviart-Thomas element space, just as the postprocessing technique in . And then we analyze the error estimates of this post-processed displacement in (div)-seminorm and -norm, which are also uniform with respect to . Finally, some numerical results are shown to demonstrate the theoretical results.
The rest of this paper is organized as follows. The reduced local discontinuous Galerkin method for nearly incompressible linear elasticity is presented in Section 2. In Section 3, we give a robust error analysis for RLDG method. A postprocessing of the displacement of RLDG method is discussed in Section 4. And in Section 5, some numerical results are included to confirm our theoretical convergence orders. In the final section, we give some concluding statements.
2. The RLDG Method for Linear Elasticity
Assume that is a bounded polygon or polyhedron. Let be the stress, let be the displacement, and let be the body force. Denote by the linearized strain tensor with , tr the trace operator, and the divergence operator. Consider linear elasticity in the stress-displacement formulation: where is the compliance tensor of the fourth order defined by Here, is the trace operation of a second-order tensor (a matrix), the usual Kronecker delta, and the positive constants and are the Lamé coefficients of the elastic material under consideration with .
To define a reduced LDG method for solving problem (1), we first introduce some notations frequently used later on. Denote the space of all symmetric tensor by . Given a bounded domain and a nonnegative integer , let be the usual Sobolev space of functions on , and let be the usual Sobolev space of functions taking values in the finite-dimensional vector space for being or . The corresponding norm and seminorm are denoted, respectively, by and . If is , we abbreviate them by and , respectively. Let be the closure of with respect to the norm . We also denote by (div the Sobolev space consisting of all functions whose divergence are square integrable. Let stands for the set of all polynomials in with the total degree no more than , and denotes the tensor or vector version of for being or , respectively.
Let be a regular family of triangulations of (cf. [22, 23]). For each , we denote by the diameter of and by the diameter of the biggest ball included in . Let be the union of all faces of the triangulation and let be the union of all interior faces of the triangulation . For a face , is set to be the diameter of . The triangulations we consider can have hanging nodes, but they have to be regular. Moreover, we restrict the ratio of the sizes of neighbor element domains. To formally state this property, we need to introduce the set defined as follows: Thus we assume that there exists a positive constant such that, for each element , This assumption forbids the situation where the mesh is indefinitely refined in only one of two adjacent subdomains. Denote by and the elementwise counterparts of and related to , respectively.
Let and be two adjacent elements of . Let be an arbitrary point of the set , and let and be the corresponding outward unit normals at that point. For a vector-valued function smooth inside each element , let us denote by the trace of on from the interior of . Then we define averages and jumps at as follows: If is on a face lying on the boundary , the above terms are defined by where is the unit outward normal vector on . We define a matrix valued jump of a vector as follows: where denote the matrix whose th component is for two vectors and .
Based on the triangulation , let The corresponding finite element spaces are given by where, for each , is a finite-dimensional symmetric tensor space of polynomials in containing , with . Let which was used to construct a family of symmetric nonconforming mixed finite elements for linear elasticity in . It is obvious that .
With former notations, we propose a reduced local discontinuous Galerkin (RLDG) method for nearly incompressible linear elasticity problem (1). Find such that where For comparison, let us recall the LDG method proposed in  in the following. Find such that where with . Formally, RLDG method (12)-(13) can be viewed as a special case of LDG method (15) with . These two methods are actually different. LDG method requires to be positive to ensure the stability, whereas RLDG method drops this requirement by choosing large enough finite element space to match in the sense that .
On the other hand, according to the definition of , the following holds: With (17), RLDG method (12)-(13) will be reduced to the symmetric nonconforming mixed finite element method designed in  by taking . Thus RLDG method can be also regarded as the discretization of the symmetric nonconforming mixed finite element method when , for there is not any continuity of finite element space across interior faces. The merit of RLDG method over the symmetric nonconforming mixed finite element method is that we do not need to solve a painful saddle problem in RLDG method. By the way, the symmetric nonconforming mixed finite element method is solved by the hybridized technique in .
Proof. Since and are both finite-dimensional, it is enough to show that if , then , . Taking in (12) and in (13) and subtracting (13) from (12), it follows that . Thus from the definition of . Then (12) is reduced to Since and (17), thus Then we can get by adopting the same argument used in the proof of Corollary 2.4 in , which ends the proof.
Next, let us consider the implementation of RLDG method (12)-(13). Let and be the bases of and , respectively. Write And denote Then RLDG method (12)-(13) has the matrix representation with , , , , and . Since is a piecewise polynomial space without any continuity across faces in , we can rearrange the basis such that is a block diagonal matrix associated with triangulation . Thus it is trivial to compute the inverse of matrix . Then the last symmetric indefinite linear system can be reduced to The new linear system is much easier to solve compared to original system, for new linear system has only unknown and is symmetric and positive definite. After obtaining by solving the new linear system, we can get from Therefore, RLDG method (12)-(13) can be implemented efficiently.
In the end of this section, we will intend to drive the primal formulation of RLDG method (12)-(13) using the argument taken in . For this, assume that in this paragraph. Define a global lifting operator by (cf. ) It holds the following inequality for lifting operator (cf. ): From (12), by integration by parts and definition of lifting operator we have Thus we have from (25) Substituting (29) into (13) and using integration by parts and definition of lifting operator , we gain the primal formulation of RLDG method (12)-(13) as follows. Find such that
3. A Priori Error Analysis
In this section, we provide an a priori error analysis for RLDG method (12)-(13). For a function with for all , let and be the usual broken -type norm and seminorm of : If is a vector-valued or tensor-valued function, the corresponding and are defined in the similar manners. For a vector or tensor , its length is or . Here the symbol denotes the double dot product operation of tensors. Throughout this paper, we use the notation “” to mean that “≤,” where is a generic positive constant independent of local element sizes and , which may take different values at different appearances.
According to the definition of , we have, for any second-order tensor , Given a bounded domain , we define two energy-type norms for in the following. For any , let By the definition of , it is obvious that Then we denote by the Hilbert inner product corresponding to energy norm . If is , we abbreviate , , and by , , and , respectively. For any , we obtain from Cauchy-Schwarz inequality For later uses, we introduce two interpolation operators. Let be the linear operator from for any to defined by in , and let be the projection operator from onto the finite element space . From in , it holds for all
Lemma 2. For all and , it follows
Next, let us study the Galerkin orthogonality of RLDG method (12)-(13). Using integration by parts, we can get from problem (1) that Subtracting (12)-(13) from (40), we obtain the following error equations:
To derive the a priori error estimates of RLDG method (12)-(13), we require as in  some regularity result for the solution of problem (1). We say the solution of problem (1) is -regular, where is a nonnegative integer whenever and there holds (cf. [4, 5, 17, 24]) for . We also assume in this paper that Moreover, introduce the usual BDM element space (cf. [25, 26]) Let be the BDM nodal interpolation operator. For simplicity, we still write and for and . Using Lemma 2, error equations (41)-(42) can be rewritten as The error estimates for interpolation operators and are summarized in the following lemma (cf. [21, 26]).
Lemma 3. For all , with a positive integer and all , one has the estimates where .
Lemma 4. For all and , it follows
Proof . Applying integration by parts, Then using (35) and triangle inequality, On the other hand, according to (44) and the definitions of trace operator and lifting operator , Since implies on , which means in . Thus by the definition of , Therefore, we can finish the proof by combining last inequality and (51).
Lemma 5. Assume that the solution of problem (1) is -regular, where is a nonnegative integer. For all , it follows
With previous preparation, we are ready to give the a priori error estimates.
Proof . Choosing in (46) and in (47) and subtracting (47) from (46), we get Then we have Applying Cauchy-Schwarz inequality and Lemma 5, it follows that is, By triangle inequality and (34), we get The proof is finished from Lemma 3.
Using the usual duality argument, we can additionally derive an optimal order error estimate between and in -norm.
Proof . Let be the solution of the auxiliary problem it follows: For is a convex bounded polygon or polyhedron, the solution of problem (66) has -regularity estimate (cf. ): Multiplying on both sides of the second equation of (66) and integrating on , we have By Lemma 2, we get Choosing in error equation (41), it holds Then it follows from the last three equalities that For the first term of the right hand side in (71), applying Cauchy-Schwarz inequality, Lemma 3, and regularity (67), it holds For the second term of the right hand side in (71), we get from Cauchy-Schwarz inequality, (34), Theorem 6, Lemma 3, and regularity (67) For the third term of the right hand side in (71), by the first equation of (66), integration by parts, and using error equation (42), we get According to Lemma 5 with regularity (67), triangle inequality, (34), Theorem 6, and Lemma 3, we have On the other hand, using Cauchy-Schwarz inequality, Lemma 3, and regularity (67), Thus we know from the last two inequalities that Combining (71)–(77) gives Finally, we can conclude the result by cancelling on both sides.
4. Postprocessing of RLDG Solution
In this section, we will focus on reconstructing an approximate displacement in the Raviart-Thomas element space via the ideas in . Assume that triangulation is conforming in this section; that is, there is no hanging nodes in . The Raviart-Thomas element space is defined by (cf. [26, 27]) With this space, the postprocessing can be described as follows. Find such that for any , where is the unit outward normal on , pointing at outside of . Since (80) are defined locally, the function can be obtained very conveniently and in parallel. To show its approximation property, we first establish a relation between the divergence of the post-processed displacement and the RLDG displacement . For this, let And denote the -projection operator from onto the finite element space . We first recall two results which have been proved in 
Now we can give the error estimate for as follows.
Proof . From (82), and the definition of , it follows that
Then by triangle inequality, and the boundedness of (cf. [22, 23]), it follows that
On the other hand, it is well known that -projection operator has the following estimate (cf. [22, 23]):
Hence, we obtain from the last two inequalities and Corollary 7 that
This completes the proof of (84).
As for (85), noting the fact that for any , we get from (83) and inverse inequality Summing over all elements in , it gives Then using triangle inequality, Therefore, we can obtain (85) from Theorem 8 and Lemma 3.
5. Numerical Results
In this section, some numerical results are shown to validate the theoretical convergence rates of RLDG method. Let , , and It is not difficult to check that the exact solution of (1) for this load is
We make a uniform triangulation on . Set for all and .
First, let us consider the convergence of RLDG method for fixed Lamé coefficient . Errors between and for are listed in Table 1, from which we can see that the convergence rates of and are and , respectively. These numerical results agree with theoretical results in Theorems 6 and 8. We also present the numerical convergence rates of postprocessing solution in Table 2 for . As Theorem 9 indicates that the convergence rates of and are and , respectively. To illustrate the robustness of RLDG method with respect to , we give numerical results for in Table 3, which is very close to the incompressible case. It can be observed from Table 3 that the convergence rates of and being again and do not deteriorate even for such large ; hence our RLDG method is locking-free.