The Weak Galerkin Finite Element Method For The Elliptic PDE Eigenvalue Problems

In recent years,the PDE eigenvalue problems have drawn more and more concentra-tion in technology and engineering area[51].Among the PDE eigenvalue problems,the elliptic PDE eigenvalue problems are of great importance.For example,the Laplacian eigenvalue problems can be used to obtain the Poincare constant of a region[68,112],and they are also useful in the spectral theory of the nonlinear PDEs[107].In the physical area,the eigenvalue problems are usually connected to the vibration,especially the reso-nance phenomenon[2,18,41]..The eigenvalue problems are also widely applied in other engineering fields,such as plasma physics in fusion experiments,astrophysics,oil reservoir simulations.Fluid flow linear stability mechanics and electronic energy band structure calculations,etc.More applications of the PDE eigenvalue problems can be found in[51].Many numerical methods have been applied to solving the eigenvalue problems,such as the finite difference method[71,89],the finite element method[5,7,92,106,25,49,35,29],the spectral method[93,69,109,87,4,11],and the discontinuous finite element method[40,141,12].However,there are still two difficulties in the numerical simulation of the eigenvalue problems.One difficulty is that the calculation cost is very high.The reason is that the eigenvalue problems are nonlinear problems.It needs to solve high dimensional linear equations many times in the iteration of the nonlinear problems,which leads to high calculation cost and large memory space.The other difficulty is to get the lower bound estimation of the eigenvalue problems.Due to the minimum-maximum principle,the conforming finite element method can only obtain the upper bounds of the eigenvalue problems.In order to get the accurate interval the eigenvalues lie in,it is also necessary to obtain the lower bounds of the eigenvalues.In this paper,we use the weak Galerkin finite element method to obtain the lower asymptotic lower bounds of the elliptic eigenvalue problems,and increase the efficiency by using acceleration algorithms.The weak Galerkin method was firstly proposed in[120]for the second order elliptic problems on polygonal or polyhedra meshes.The main idea of the weak Galerkin method is to use the discontinuous polynomials as basis,and replace the classical derivative oper-ators by the specifically defined weak derivative operators.Since the finite element space of the weak Galerkin method consists of discontinuous polynomials,it does not need to consider the continuity of the cell basis functions when constructing the weak Galerkin finite element space,i.e,the basis function of the weak Galerkin finite element space can be defined on each element,and the different elements do not affect each other.This gives flexibility to the weak Galerkin finite element method.Due to the lack of continuity condi-tions,the weak finite element method can be applied to general polygonal meshes and can be easily extended to three-dimensional situations.For discontinuous piecewise polyno-mials,the weak derivative operators defined by distribution is another important feature of the weak Galerkin finite element method.Weak derivative operators characterize the combined effect of the values of the interior and boundary functions on the derivative val-ues better,and make numerical methods have better approximation.The weak Galerkin method has been applied to many different types of PDEs,such as the biharmonic equa-tion[98,101,146],the Stokes equation[122,143],the Brinkman equation[97,123,142],and the Maxwell equation[102].The accelerating algorithms used in this paper include the two-grid method,the two-space method,and the shift-inverse powered method.The two-grid method in[131]is first used to solve semilinear second-order elliptic equaiton,and then it is applied to other types of nonlinear partial differential equations[132]and the eigenvalue problems[133].The main idea of the two-grid method is to transform the large-scale eigenvalue problem on a fine mesh into an eigenvalues on a coarse mesh and a linear problem on the fine mesh,while the coarse mesh step size H and fine mesh step h are chosen to maintain the same precision.For example,for the Laplacian eigenvalue problems,the mesh size ratio can be chosen as H =(?)to reduce the cost of computation.The two grid method is also used in other types of eigenvalue problems[134,136],and further developed the multigrid method[36,64,128].This paper contains two parts.In the first part,we apply the weak Galerkin finite element method to the elliptic PDE eigenvalue problems and give a convergence analysis.We discuss the properties of the weak Galerkin finite element method in the eigenvalue problems,construct a general analysis framework for the asymptotic lower bound estima-tion,and give detailed analysis for the Laplacian eigenvalue problems and the biharmonic eigenvalue problems.We also study the post-processing techniques based on interpolation,and propose an algorithm for the weak Galerkin finite element method which get the upper bound bounds of the eigenvalue problems without solving auxiliary problems.In the sec-ond part we mainly focus on the acceleration of the weak Galerkin finite element method.We use the two-grid method,the two-space method and the shift-inverse powered method to accelerate the weak Galerkin finite element method for the eigenvalue problems.We also study how to improve the efficiency by acceleration algorithm while the asymptotic lower bound property of the weak Galerkin finite element method is maintained.The paper is structured as follows.The first part contains Chapter 2 and Chapter 3.We study the general framework of the weak Galerkin finite element method for the elliptic PDE eigenvalue problems,and apply the framework to the Laplaican eigenvalue problem and the biharmonic eigenvalue problem.In Chapter 2,we consider the abstract eiganvalue problem as follows a(u,v)= ?b(u,v),(?)v?Vc,(7)aw(Uh,vh)=?hb(uh,vh),(?)vh?Vh,(8)wehre Vc denotes the Hilbert space for the exact solution,Vh denotes the finite dimen-sional Hilbert space for the numerical solution.Vc and Vh are subspace of Hilbert space V,but Vh(?)Vc.For this abstract problem,we propose the following 7 assumptions.Assumption(A1)a(.,.),aw(.,.),b(.,.)are symmetric bilinear forms.For any v?Vc and vh?Vh,we have where ?c and ?(h)are positive.Assumption(A2)The solutio-n operator of a(.,.)and aw(.,.),K and Kh are compact.Assumption(A3)There exists a bounded linear operator Qh;V ?Vh salisfying Assumption(A4)When h?0,we have eh,??0 and ?h,??(h)-1?0.Assumption(A5)When h?0,we have eh,?'?0 and ?h,??'(h)-1?0.Assumption(A6)For any ???(K)and u ?R(E?(K)),when h?0,we have ?h,u?0.Assumption(A7)Suppose(?,u)is the exact eigenpair and(?h,uh)is the numerical eigen-pair,and we have ?h,u??h||u-uh||X2.The definition of symbols K,Kh,eh,?,?h,?,eh,?',?h,?',?(K),R(E?(K)),and ?h,? are listed in Section 2,Chapter 2.Based on these assumptions,from the spectral theory and the expansion of the eigenvalue we have the following error estimates and lower bound estimates.Theorem Under Assumption(A1)-(A6))suppose ? is the exact eigenvalue with multiplic-itym,{uj}j=1m is a set of eigenvectors corresponding to ?.When h is sufficiently small,there existm numerical eigenvalues{?h,j}j=1m such that for any j=1,…,m,we haveTheorem Under Assumptions(A1),(A3),and(A7),suppose(?,^u)is the exact eigen-pair,(?h,uh)is the numerical eigenpair,then we have?>?h.Next we apply the framework to Laplaciln eigenvalue problem and the biharmonic eigenvalue problem.Consider the Laplacimn eigenvalue problem From the above framework,we construct the corresponding weak Galerkin scheme and give the error estimations and lower bound estimation.Theorem Suppose ? is the eigenvalue of the Laplacian eigenvalue problem with multiplici-ty m,R(E?(K))(?)Hk+1(?)is the correspondingm dimensional eigenspace.Suppose {?h,j}j=1,are numerical eigenvalues converging to ?,and {uh,j}j=1m is a basis of the eigenspace R(E?,h(Kh)).Then when h is sufficiently small,for anyj = 1,…,m we haveTheorem Suppose(?,u)is an eigenpair of the Laplacian eigenvalue probleml and(?h,uh)is a numerical eigenpair converging to(?,u).When ?(h)<<1,and is sufficiently small,we have? ? ?h.We also apply this method to the biharmonic eigenvalue problem Using the similar techniques,we have the following error estimate and lower bound esti-mate.Theorem Suppose ? is the eigenvalue of the biharmonic eigenvalue problem with multi-plicity m,and R(E?(K))(?)Hk+2(?)is the corresponding m dimensional eigenspace.Sup-pose {?h,j}j=1m are numerical eigenvalues converging to ?,and {uh,j}j=1m is a basis of the eigenspace R(E?,h(Kh)).Then when h is sufficiently small,for any j = 1,…,m we haveTheorem Suppose(?,u)is an eigenpair of the biharmonic eigenvalue problem,and(?h,uh)is a numerical eigenpair converging to(?,u).Then when ?(h)<<1,and h is sufficiently small,we have?>?h.In Chapter 3,we research the upper bound problem of the eigenvalue problems.In this chapter,we still consider the Laplacian eigenvalue problem.In Chapter 2 we prove that the weak Galerkin finite element method provides the lower bounds of the Laplacian eigenvalue problems.In this chapter,we get the upper bounds by using interpolation post-processing method,and it does not need to solve any auxiliary problem.The algorithm is as follows.Algorithm 1 Step 1.Find ?h?R,uh?Vh such that bw(uh,uh)=1 satisfying Step 2.Calculate uh = ?huh.Stpe 3.Calculate the Rayleigh quotientThe eigenvalue produced by the interpolation algorithmAh is an upper bound of the exact eigenvalue ?.The numerical eigenvalue ?h and ?h have the same accuracy.The in-terpolation algorithm has the following error estimate.Theorem Suppose the exact eigenfunction of the Laplacian eigenvalue problemuj?Hk+1(?)is not a polynomial with degree k.Suppose ?j,h is the j-th eigenvalue of the interpolation algorithm,then there exists exact eigenvalue ?j such that when h is sufficiently small,the following estimate holds In Chapters 2 and 3,we also design numerical experiments to verify the theoretical estimates.The second part includes Chapter 4 and Chapter 5.In this part,we use the two-grid method,the two-space method and the shift-inverse powered method to accelerate the weak Galerkin finite element method in order to improve the efficiency.In Chapter 4,we study the two-grid methods and the two-space method.In the two-grid method,two sets of grids are necessary.One set is a coarse mesh with a larger grid step size,and the other set is a fine mesh with a smaller step size.The goal of the two-grid method is to get the numerical solution of the weak Galerkin finite element method for the eigenvalue problem on a fine grid efficiently.However,since the fine grid has a small step size,the number of elements is large,so the number of degrees of freedom is large.The eigenvalue problem of partial differential equations often leads to a matrix generalized eigenvalue problem.The matrix generalized eigenvalue problems are c.ompletely nonlinear problems,which require many iterations.The number of degrees of freedom on the fine grid is large,and the dimension of the matrix dimension is high.Solving the eigenvalue problem directly on the fine mesh is computationally expensive and difficult to solve the eigenvalue problem efficiently.The two-grid method transforms the eigenvalue problem on a fine grid into an eigen-value problem on a coarse grid and a linear equations problem on a fine grid.The number of degrees of freedom on a coarse grid is small,so the resulting generalized eigenvalue problem is easier to solve.On a fine grid,only the linear equations problem needs to be solved.Thus,compared with the generalized eigenvalue problem on a fine grid,the computational complexity is greatly reduced.Therefore,the efficient can be effectively improved.For the Laplacian eigenvalue problem,the two-grid algorithm for the weak Galerkin finite element method is as follows:Algorithm 2 Stpe 1:Generate a coarse resh TH on ?,and solve the eigenvalue problem on TH:Find(?H,uH)?R×VH with bw(uH,uH)=1,such that Step 2:Refine the coarse mesh TH to get a fine mesh Th.Solve the linear problem on Th:Find uh ?Vh such that Step 3:Calculate the Rayleigh quotient of uh Finally,we get the approximate eigenpair(?h,uh).By selecting the mesh step of the coarse grid and the fine grid properly,the solution efficiency can be improved without loss of accuracy,and the eigenvalue asymptotic lower bound estimate can also be maintained.Theorem Suppose(?h,uh)is the eigenpair of the two-grid method,h<H and the Lapla-cian eigenvalue problem has Hk-1(?)-regularity.Then there exists exact eigenpair(?,u)such that where k = min{2k-2?,k+?-?}.Theroem Under the assumptions of the previous theorem,suppose k = min{2k-2?,k +?-?},and ? is a positive number.When H2k ? Ch2k-?,we have?-?h?0,where H and h are sufficiently small.In Chapter 4,we study the two-space method.The main idea of the two-space method is similar to the two-grid method,both of which transform the large-scale eigenvalue problem into a small-scale eigenvalue problem and a large-scale linear problem.The two-Brid method is to obtain two weak Galerkin finite element spaces by using different meshes.The two-space method is to obtain two weak Galerkin finite element spaces by using different degrees of polynomials.With less degrees of freedom in the lower order spaces,the computation of the eigenvalue problem is also relatively small,and there are more degrees of freedom in the high order spaces,which can achieve higher approximation accuracy.In the two-space method,by solving the eigenvalue problem in the low-order space and the linear problem in the high-order space,the calculation cost can be reduced without loss of accuracy,and the efficiency can be improved.In Chapter 5,we further accelerate the weak Galerkin finite element method by the shift-inverse powered method.Similar to the two-grid method,the main idea of the shift-inverse powered method is to transform the eigenvalue problem on a fine grid into an eigenvalue problem on a coarse grid and a linear problem on the fine grid.The difference in the algorithm is as follows.In the two-grid method,the linear problem in Step 2 is:Find uh?Vh such that In the shift-inverse powered method,the linear problem is replaced by:Find uh ?Vh such thatBy modifying the algorithm,the requirement of mesh step ratio is weaker.For the Laplacian eigenvalue problem,the two-grid method needs H?(?)to get the optimal error estimate,and the shift-inverse powered method only needs H ?(?).For example,when the fine mesh steph = 1/256,the two-grid method needs to solve the eigenvalue problem on a coarse mesh with mesh step H = 1/16,and the shift-inverse powered method only needs to solve the eigenvalue problem on a coarse mesh with mesh step H = 1/4,which further reduce the computational cost and improve the efficiency.In Chapter 4 and Chapter 5,we present some numerical experiments to verify the efficiency of the algorithms.