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In the numerical solution of
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ho ...
, a topic in mathematics, the spectral element method (SEM) is a formulation of the
finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
(FEM) that uses high degree
piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. ...
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s as basis functions. The spectral element method was introduced in a 1984 paper by A. T. Patera. Although Patera is credited with development of the method, his work was a rediscovery of an existing method (see Development History)


Discussion

The
spectral method Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis function ...
expands the solution in
trigonometric Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
series, a chief advantage being that the resulting method is of a very high order. This approach relies on the fact that
trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The ...
s are an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
for L^2(\Omega). The spectral element method chooses instead a high degree piecewise polynomial basis functions, also achieving a very high order of accuracy. Such polynomials are usually orthogonal
Chebyshev polynomial The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...
s or very high order
Lagrange polynomials In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' ...
over non-uniformly spaced nodes. In SEM computational error decreases exponentially as the order of approximating polynomial increases, therefore a fast convergence of solution to the exact solution is realized with fewer degrees of freedom of the structure in comparison with FEM. In
structural health monitoring Structural health monitoring (SHM) involves the observation and analysis of a system over time using periodically sampled response measurements to monitor changes to the material and geometric properties of engineering structures such as bridges a ...
, FEM can be used for detecting large flaws in a structure, but as the size of the flaw is reduced there is a need to use a high-frequency wave. In order to simulate the propagation of a high-frequency wave, the FEM mesh required is very fine resulting in increased computational time. On the other hand, SEM provides good accuracy with fewer degrees of freedom. Non-uniformity of nodes helps to make the mass matrix diagonal, which saves time and memory and is also useful for adopting a central difference method (CDM). The disadvantages of SEM include difficulty in modeling complex geometry, compared to the flexibility of FEM. Although the method can be applied with a modal piecewise orthogonal polynomial basis, it is most often implemented with a nodal tensor product Lagrange basis.Karniadakis, G. and Sherwin, S.: Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford Univ. Press, (2013), The method gains its efficiency by placing the nodal points at the Legendre-Gauss-Lobatto (LGL) points and performing the Galerkin method integrations with a reduced Gauss-Lobatto quadrature using the same nodes. With this combination, simplifications result such that mass lumping occurs at all nodes and a collocation procedure results at interior points. The most popular applications of the method are in computational fluid dynamics and modeling seismic wave propagation.


A-priori error estimate

The classic analysis of
Galerkin method In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete probl ...
s and
Céa's lemma Céa's lemma is a lemma in mathematics. Introduced by Jean Céa in his Ph.D. dissertation, it is an important tool for proving error estimates for the finite element method applied to elliptic partial differential equations. Lemma statement Let V ...
holds here and it can be shown that, if u is the solution of the weak equation, u_N is the approximate solution and u \in H^(\Omega): :\, u-u_N\, _ \leqq C_s N^ \, u \, _ where N is related to the discretization of the domain (ie. element length), C_s is independent from N, and s is no larger than the degree of the piecewise polynomial basis. Similar results can be obtained to bound the error in stronger topologies. If k \leq s+1 \, u-u_N \, _ \leq C_ N^ \, u\, _ As we increase N, we can also increase the degree of the basis functions. In this case, if u is an
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
: :\, u-u_N\, _ \leqq C \exp( - \gamma N ) where \gamma depends only on u. The Hybrid-Collocation-Galerkin possesses some superconvergence properties. The LGL form of SEM is equivalent,Young, L.C., “Orthogonal Collocation Revisited,” Comp. Methods in Appl. Mech. and Engr. 345 (1) 1033-1076 (Mar. 2019)
doi.org/10.1016/j.cma.2018.10.019
/ref> so it achieves the same superconvergence properties.


Development History

Development of the most popular LGL form of the method is normally attributed to Maday and Patera. However, it was developed more than a decade earlier. First, there is the Hybrid-Collocation-Galerkin method (HCGM),Wheeler, M.F.: “A C0-Collocation-Finite Element Method for Two-Point Boundary Value and One Space Dimension Parabolic Problems,” SIAM J. Numer. Anal., 14, 1, 71-90 (1977) which applies collocation at the interior Lobatto points and uses a Galerkin-like integral procedure at element interfaces. The Lobatto-Galerkin method described by Young is identical to SEM, while the HCGM is equivalent to these methods. This earlier work is ignored in the spectral literature.


Related methods

* G-NI or SEM-NI are the most used spectral methods. The Galerkin formulation of spectral methods or spectral element methods, for G-NI or SEM-NI respectively, is modified and Gauss-Lobatto integration is used instead of integrals in the definition of the bilinear form a(\cdot,\cdot) and in the functional F. Their convergence is a consequence of Strang's lemma. *SEM is a Galerkin based FEM (finite element method) with Lagrange basis (shape) functions and reduced numerical integration by Lobatto quadrature using the same nodes. *The pseudospectral method,
orthogonal collocation In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate solutions (usually ...
, differential quadrature method, and G-NI are different names for the same method. These methods employ global rather than piecewise polynomial basis functions. The extension to a piecewise FEM or SEM basis is almost trivial. * The spectral element method uses a
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes ...
space spanned by nodal basis functions associated with Gauss–Lobatto points. In contrast, the p-version finite element method spans a space of high order polynomials by nodeless basis functions, chosen approximately orthogonal for
numerical stability In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorit ...
. Since not all interior basis functions need to be present, the p-version finite element method can create a space that contains all polynomials up to a given degree with fewer degrees of freedom. However, some speedup techniques possible in spectral methods due to their tensor-product character are no longer available. The name ''p-version'' means that accuracy is increased by increasing the order of the approximating polynomials (thus, ''p'') rather than decreasing the mesh size, ''h''. * The ''hp'' finite element method (
hp-FEM hp-FEM is a general version of the finite element method (FEM), a numerical method for solving partial differential equations based on piecewise-polynomial approximations that employs elements of variable size ''(h)'' and polynomial degree '' ...
) combines the advantages of the ''h'' and ''p'' refinements to obtain exponential convergence rates.P. Šolín, K. Segeth, I. Doležel: Higher-order finite element methods, Chapman & Hall/CRC Press, 2003.


Notes

{{DEFAULTSORT:Spectral Element Method Numerical differential equations Partial differential equations Computational fluid dynamics Finite element method