Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical ModelsView this Special Issue
A Class of Spectral Element Methods and Its A Priori/A Posteriori Error Estimates for 2nd-Order Elliptic Eigenvalue Problems
This paper discusses spectral and spectral element methods with Legendre-Gauss-Lobatto nodal basis for general 2nd-order elliptic eigenvalue problems. The special work of this paper is as follows. (1) We prove a priori and a posteriori error estimates for spectral and spectral element methods. (2) We compare between spectral methods, spectral element methods, finite element methods and their derived -version, -version, and -version methods from accuracy, degree of freedom, and stability and verify that spectral methods and spectral element methods are highly efficient computational methods.
As we know, finite element methods are local numerical methods for partial differential equations and particularly well suitable for problems in complex geometries, whereas spectral methods can provide a superior accuracy, at the expense of domain flexibility. Spectral element methods combine the advantages of the above methods (see ). So far, spectral and spectral element methods are widely applied to boundary value problems (see [1, 2]), as well as applied to symmetric eigenvalue problems (see ). However, it is still a new subject to apply them to nonsymmetric elliptic eigenvalue problems.
A posteriorii error estimates and highly efficient computational methods for finite elements of eigenvalue problems are the subjects focused on by the academia these years; see [3–16], and among them, for nonsymmetric 2nd-order elliptic eigenvalue problems, [5, 15] provide a posteriori error estimates and adaptive algorithms,  the function value recovery techniques and [8, 10] two-level discretization schemes.
Based on the work mentioned above, this paper shall further apply spectral and spectral element methods to nonsymmetric elliptic eigenvalue problems. This paper will mainly perform the following work.(1)We prove a priori and a posteriori error estimates of spectral and spectral element methods with Legendre-Gauss-Lobatto nodal basis, respectively, for the general 2nd-order elliptic eigenvalue problems.(2)We compare between spectral methods, spectral element methods with Legendre-Gauss-Lobatto nodal basis, finite element methods, and their derived -version, -version, and -version methods from accuracy, degree of freedom, and stability and verify that spectral methods and spectral element methods are highly efficient computational methods for nonsymmetric 2nd-order elliptic eigenvalue problems.
This paper is organized as follows. Section 2 introduces basic knowledge of second elliptic eigenvalue problems. Sections 3 and 4 are devoted to a priori and a posteriori error estimates of spectral and spectral element methods, respectively. In Section 5, some numerical experiments are performed by the methods mentioned above.
Consider the 2nd-order elliptic boundary value problem where is a bounded domain, and are a real-valued vector function and a real-valued function, respectively, and is a positive scalar function with .
We denote the complex Sobolev spaces with norm by , . Let and be a inner product and a norm in the complex space , respectively.
In this paper, denotes a generic positive constant independent of the polynomial degrees and mesh scales, which may not be the same at different occurrences.
Define the bilinear form as follows:
We assume that , , and are bounded functions on , namely . Further more, we assume that exists and satisfies
Under these assumptions, the bilinear form is continuous in and -elliptic; that is, there exist two constants independent of such that
The corresponding variational formulation of (1) is given as follows: find , such that
The adjoint problem of (5) is as follows: find , such that
We assume that is a regular rectangle (resp. cuboid) or simplex partition of the domain and satisfies . We associate with the partition a polynomial degree vector , where is the polynomial degree in . Let be the diameter of the element , and let .
We define spectral and spectral element spaces as follows: where and are polynomial spaces of degree (resp. degree in every direction) in and degree (resp. degree in every direction) in the element , respectively.
The spectral approximation of (5) is as follows: find , such that
The spectral element approximation of (5) is as follows: find , such that
Define the interpolation operators as the interpolations in the element and the domain , respectively, with the tensorial Legendre-Gauss-Lobatto (LGL) interpolation nodes.
Define the interpolation operator
We quote from  (see (5.8.27) therein) the interpolation estimates for spectral and spectral element methods with LGL Nodal-basis as follows.
For all ,
For all ,
Particularly, if , then we have
Using Aubin-Nitsche technique, we deduce from the regularity estimate (8) and the estimates (18)–(20) the priori estimates of boundary value problem (5) for spectral and spectral element methods; that is,
3. Spectral and Spectral-Element Approximations and Error Estimates for Eigenvalue Problems
3.1. Spectral and Spectral-Element Approximations for Eigenvalue Problems
Consider the following eigenvalue problem corresponding to the boundary value problem (1):
The variational formation of (23) is given by the following: find , such that
The spectral approximation scheme of (24) is given by the following: find , such that
The spectral element approximation scheme of (24) is given by the following: find , such that
Find , such that
Find , such that The adjoint problem of the eigenvalue problem (23) is where .
The variational formation of (30) is given by the following: find , such that
The spectral element approximation scheme of (31) is given by the following: find , such that
Let be an eigenvalue of (23). There exists a smallest integer , called the ascent of , such that . is called the algebraic multiplicity of . The functions in are called generalized eigenfunctions of corresponding to . Likewise the ascent, algebraic multiplicity and generalized eigenfunctions of , and can be defined.
Let be an eigenvalue of (23) with the algebraic multiplicity and the ascent . Assume ; then there are eigenvalues () of (26) which converge to . Let be the space spanned by all generalized eigenfunctions corresponding to of , and let be the space spanned by all generalized eigenfunctions corresponding to all eigenvalues of that converge to .
3.2. A Priori Error Estimates
We will analyze a prior error estimates for spectral element methods which are suitable for spectral methods with mesh fineness not considered.
Assume that and are two closed subspace in .
Denote We say that is the gap between and .
We give the following four lemmas from Theorem 8.1–8.4 in , which are applications to spectral element methods.
Lemma 1. Assume . For small enough and big enough , there holds
Lemma 2. Assume ; then
Lemma 3. Assume that ; then there holds Since is a finite dimensional space, there exists a direct-sum decomposition . We define the operator as a projection along from to .
Lemma 4. Assume . Let be an eigenvalue of and . satisfies and , where is a positive integer. Then, for every integer , there holds
We assume that in this section, for the sake of simplicity, .
Theorem 5. If and , then there holds the following error estimates: Let , and let , for some . Then, for every integer , there exists a function , such that and where , .
Proof. We derive from the error estimate (20) that By (14), Analogically, Plugging the two inequalities above into (36), (38), and (39) yields (42), (41), and (43), respectively. We find from (37) that combining with (45) and (46) yields (40).
Supposing that , is a regular set of , and is a closed Jordan curve enclosing .
Define the spectral projection operators
Lemma 6. If , then there holds that , is uniformly bounded with and , and
Theorem 7. Under the assumptions of Theorem 5, further assume that the ascent of is . Let be an eigenpair of (26) with ; then there exists an eigenpair of (24), such that
where and .
Let be an eigenpair of (24). If is an eigenvalue of (26) convergence to , then there exists , such that (51)–(53) hold.
Proof. We deduce (53) immediately from (41). We derive from (22) and (7) that
thus,. Taking and by virtue of , Lemma 6 and (22), we have
from which follows
which is (52). By direct calculation, we have
Plugging (20), (52), and (53) into (57) yields (51).
If is an eigenpair of (24), let ; by the same argument we can prove (51) and (52).
4. A Posteriori Error Estimates
Based on , we will discuss a posteriori error estimates. We further assume that , the partition is -shape regular, and the polynomial degree of neighboring elements are comparable; that is, there exists , such that for all , ,
We refer to the -clément interpolation estimates given by [20, 21] (see theorems 2.2 and 2.3, respectively), which generalize the well-known clément type interpolation operators studied in  and  to the hp context.
Lemma 8. Assume that the partition is -shape regular and the polynomial distribution is comparable. Then there exists a positive constant and the clément operator , such that
where is the length of the edge and , where are elements sharing the edge and are patches covering and with a few layers, respectively.
Define interval and weight function . Denote the reference square and triangle element by and , respectively. Define weight function .
The following three lemmas are given by . Lemmas 9–10 provide the polynomial inverse estimates in standard interval and element, while Lemma 11 provides a result for the extension from an edge to the element.
Lemma 9. Let . Then there exists , such that for all and all univariate polynomials of degree ,
Lemma 10. Let . Then there exist , such that for all and all polynomials of degree -,
Lemma 11. Let , ; then there exists such that for all , and all univariate polynomials of degree , there exists an extension and holds
It is easy to know that the three lemmas above hold for complex-valued polynomials.
Let , and be the interpolations of , , and in with the polynomial degree (resp. degree in every direction), respectively, or the -projection on the space of polynomials with degree . For convenient argument, here and hereafter we assume that and are the eigenpairs of the eigenvalue problem (24) and its adjoint problem (31), respectively. and are the solutions of the corresponding spectral element approximations (26) and (32), respectively.
Denote Define the local error indicators Their first terms are the weighted element internal residuals given by Their second terms are the weighted element boundary residuals given by where we denote the jump of the normal derivatives of and across the edges by and , respectively. is the length of edge . The weight functions and are scaled transformations of the weight functions and ; that is, if is the element map for element and is the image of the edge under , then where we choose , such that We define the global error indicators as follows:
Theorem 12. Let . Then there exists a constant independent of , and , such that
Proof. We denote , where is -clément operator given by Lemma 8. We derive from -elliptic of that
which together with
and using Cauchy-Schwartz inequality, the -clément interpolation estimates in Lemma 8 then yield
Using scaled transformation and setting in (61) and (62), we get and ; then this proof concludes.
For the adjoint eigenvalue problem, we still have the following.
Theorem 13. Let . Then there exists a constant independent of , and , such that
Lemma 14. Let . Then there exists a constant independent of , and , such that
Proof. We denote with and extend to by on ; then
We consider the semi norm for . Using the polynomial inverse estimates (62)-(63) in Lemma 10, by transformation between the reference element and , we find for that
Note that (62) is applicable since implies ; thus, we set in (62); then the third inequality above holds.
Since , we obtain To obtain an upper bound in the case of , we use the polynomial inverse estimate (62) in Lemma 10; for , we derive from (62) that Setting , We obtain the desired result immediately from (83) and (85).
In order to obtain a local upper bound for the error indicator , we consider the edge residual term . we introduce the set
Lemma 15. Let . Then there exists a constant independent of , and , such that
Proof. We will use weight functions on edge and a suitable extension operator. For a given element with edge , we choose the element so that . Denote ; we construct a function with as follows.
Let ( is defined by (71)). Using Lemma 11, we extend to , where the polynomial corresponds to . Define and as the affine transformation of in ; Thus, is a piecewise -function. From (64), we know vanishes on ; Therefore, . It is trivial to extend to , such that in . We find Therefore, We consider the case of first. Using the affine equivalence and (65)-(66) in Lemma 11, we obtain the upper bounds for and as follows: It follows from (89)-(90) that By the definition of and setting in Lemma 14, we get by the triangle inequality Combining the three inequalities above and summing, we have
Setting in the above inequality yields the assertion for . For the case of , we set , use (62) in Lemma 10 to get , and find the desired result.
Theorem 16. Let . Then there exists a constant independent of , and , such that
Similarly, we have Theorem 17.
Theorem 17. Let . Then there exists a constant independent of , and , such that