## Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models

View this Special IssueResearch Article | Open Access

# A Class of Spectral Element Methods and Its A Priori/A Posteriori Error Estimates for 2nd-Order Elliptic Eigenvalue Problems

**Academic Editor:**Rasajit Bera

#### Abstract

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.

#### 1. Introduction

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 [1]). 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 [3]). 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 [3â€“16], and among them, for nonsymmetric 2nd-order elliptic eigenvalue problems, [5, 15] provide a posteriori error estimates and adaptive algorithms, [9] 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.

#### 2. Preliminaries

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

As the general 2nd-order elliptic boundary value problems, we assume that the regularity estimates for problem (5) and its adjoint problem (6) hold, respectively. Namely

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

We assume that and derive from Lax-Milgram theorem that the variational formations (5), (6), (10), and (11) have a unique solution, respectively.

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 [2] (see (5.8.27) therein) the interpolation estimates for spectral and spectral element methods with LGL Nodal-basis as follows.

For all ,

For all ,

We assume that the solution of boundary value problem (5) , that and are the solutions of (10) and (11), respectively; then we derive from CÃ©a lemma and the interpolation estimates that where .

Particularly, if , then we have

Note that (18) is also suited to spectral methods with modal basis (see [1, 2]).

Using Aubin-Nitsche technique, we deduce from the regularity estimate (8) and the estimates (18)â€“(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

Define the solution operators , and by Obviously (see [17]), the equivalent operator forms for (24) and (26) are the following.

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

We can likewise define the equivalent operator forms for the eigenvalue problems (31) and (32) as

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 .

In view of adjoint problems (31) and (32), the definitions of and are analogous to and . Let , and let .

Note that when , both (24) and (26) are symmetric. Thus, the ascent of , and the ascent of .

##### 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 .

Denote

We give the following four lemmas from Theorem 8.1â€“8.4 in [17], 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 .

Denote

Define the spectral projection operators

We give the following lemma by referring to [18, 19] (see proposition 5.3 in [18] and theorem in [19]).

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)â€“(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 [20], 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 -clÃ©ment interpolation estimates given by [20, 21] (see theorems 2.2 and 2.3, respectively), which generalize the well-known clÃ©ment type interpolation operators studied in [22] and [23] 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 clÃ©ment 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 [20]. Lemmas 9â€“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 -clÃ©ment operator given by Lemma 8. We derive from -elliptic of that
Therefore,
which together with
and using Cauchy-Schwartz inequality, the -clÃ©ment 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