Comput. Mech. (2006) DOI 10.1007/s00466-006-0048-7
O R I G I NA L PA P E R
Guillermo Hauke · Mohamed H. Doweidar Daniel Fuster · Antonio Gómez · Javier Sayas
Application of variational a-posteriori multiscale error estimation to higher-order elements
Received: 31 October 2005 /Accepted: 09 February 2006 © Springer-Verlag 2006
Abstract An explicit a-posteriori error estimator based on the variational multiscale method is extended to higher-order elements. The technique is based on a recently derived explicit formula of the fine-scale Green’s function for higherorder elements. For the class of element-edge exact methods, the technique is able to predict the error exactly in any desired norm. It is shown that for elements of order k, the exact error depends on the k − 1 derivative of the residual. The technique is applied to one-dimensional examples of fluid transport computed with stabilized methods. Keywords A posteriori error estimation · Advection-diffusion equation · Hyperbolic flows · Fluid mechanics · Fluid dynamics · Stabilized methods · Variational multiscale method 1 Introduction The variational multiscale method [17,18] offers a fresh theoretical framework to study and develop a-posteriori error estimators [1]. This idea has been initially explored in [13] to identify an explicit a-posteriori error estimator for stabilized solutions, especially well suited for convection-dominated flows. Practical applications show that the adaptive meshes generated with this technique are of high quality. The above paper was followed by [14,12], where the proper intrinsic scales for a posteriori error estimation were derived for the class of element-edge exact methods, i.e. methods characterized by solutions which are exact along the element boundaries. Clearly, for methods where the error is zero along the element edges, the error propagation is stopped at the intereleG. Hauke (B) · M. H. Doweidar · D. Fuster · A. Gómez Departamento de Mecánica de Fluidos, Centro Politécnico Superior, C/Maria de Luna 3, 50018 Zaragoza, Spain E-mail:
[email protected] J. Sayas Departamento de Matemática Aplicada, Centro Politécnico Superior, C/Maria de Luna 3, 50018 Zaragoza, Spain
ment boundaries, and the error remains confined within each element. The operator that distributes the error in a numerical solution is the fine-scale Green’s function, which takes into consideration both, the error propagation and the projection of the exact solution into the finite element space. Furthermore, Hughes and Sangalli [19] show that in the one-dimensional case and for methods based on the H01 projector, although the Green’s function is a global function, the fine-scale Green’s function is local and confined within the elements. In the multi-dimensional setting and for advectiondominated flows (which corresponds to the hyperbolic limit) [19] also shows that the above attributes approximately hold for methods based on the H01 projector. Thus, the work here relies of these findings. One of the traits of this framework is that it gives the recipe to calculate the error constants in the desired error norm. In this context, it was also shown that the classical intrinsic time-scale parameter [6,9,17] carries error information in the L 1 norm as a function of the L ∞ residual norm. Since the error is propagated according to the fine-scale Green’s function, as the nature of this function is better understood, the above mentioned results can be extended to a wider class of problems. For instance, the work by Hughes and Sangalli [19] develops explicit formulas for higher-order finescale Green’s functions, which are exploited in this paper to extend the previous work on variational multiscale a-posteriori error estimation to higher-order elements.
2 The variational multiscale approach to error estimation 2.1 The abstract problem Consider a spatial domain with boundary . The strong form of the boundary-value problem consists of finding u : → R such that for the given essential boundary condition g : g → R, the natural boundary condition h : h → R, and forcing function f : → R, f ∈ L 2 (if h = ∅, f ∈ H −1 ), the following equations are satisfied
Guillermo Hauke et al.
Lu = f in , u = g on g , (1) B u = h on h , where L is a second-order differential operator and B an operator acting on the boundary, emanating from integration-byparts of the weak form. In what follows the analysis will be performed for the one-dimensional case.
2.2 The error estimation paradigm The variational multiscale method [17] introduces a splitting on the weak solution of (1), u ∈ S ⊂ H 1 , into u, ¯ the resolved scales or finite element solution and u , the unresolved scales or error, i.e. u = u¯ + u . (2) ¯ Typically, if u¯ belongs to the finite element space S , then u ∈ S with S = S \ S¯ . Thus, given a finite element space S¯ with elements e , e = 1, . . . , n el , and numerical solution u¯ ∈ S¯ , the error u ∈ S can be shown to obey [18,13] u (x) = − g (x, y) (Lu¯ − f )(y) d y y
−
g (x, y) ([[Bu]])(y) ¯ d y
y
−
g (x, y) (Bu¯ − h)(y) d y ,
(3)
h y where g (x, y) ∈ S × S is fine-scale Green’s function [17– = ∪ne el e are the element interiors, = ∪ne el e \ 19],
are the inter-element boundaries, and [[·]] is the jump operator [16,18]. Note that only in one-dimensional problems g (x, y) ∈ H 1 × H 1 . The above paradigm could be directly exploited to compute the error, if the fine-scale Green’s function were readily available.
2.3 Element Green’s functions The element Green’s functions are solutions of Green’s function problems restricted to one element. Under appropriate conditions, element Green’s functions represent exactly the fine-scale Green’s function. Let us denote the element Green’s function associated with the finite element functional space spanned by polynomials of order k as ge(k) . In particular, for linear finite element spaces, the element Green’s function ge(1) (x, y) satisfies within each element the boundary value problem (1) Lge = δ y in e , (1) ge = 0 on e , where δ y (x) = δ(x − y) represents the mass Dirac delta distribution.
3 Quadratic elements Recently, Hughes and Sangalli [19] have explored the nature of the fine-scale Green’s function. One of the basic findings is that the structure of the fine-scale Green’s function depends, not only on the solution functional space, but on the scale decomposition or the projection between u and u. ¯ The above paper also derives an explicit formula of the fine-scale Green’s function for higher-order elements. For the optimal H01 projection, which selects the nodal interpolant as the numerical solution, i.e. the case that occupies this paper, the fine-scale Green’s function becomes an element Green’s function, its expression for quadratic elements being ge(2) (x, y) = ge(1) (x, y) (1) (1) e ge (x, y) dx e ge (x, y) d y . (5) − (1) e e ge (x, y) dx d y
2.2.1 The smooth error estimator For certain classes of problems and methods, the error has a predominantly elementwise local behavior. For these problems, it can be safely assumed that the finite element solution u¯ is practically exact on e , that is, the error u = 0 on e . Furthermore, this implies that g vanishes on e . Then, the above a-posteriori error estimator (3) simplifies to u (x) = − g (x, y) (Lu¯ − f )(y) d y on e , (4) ey
where clearly the local character of the error can be observed [13]. Examples where this paradigm (4) is exact are one-dimensional linear problems solved with nodally-exact multiscale methods and one-dimensional Poisson problems solved with the Galerkin method.
Remarks (i) The quadratic element Green’s function has the symmetry property ge(2) (x, y) dx = ge(2) (x, y) d y = 0 (6) e
e
that is, the zeroth order moment with respect to any of its arguments is zero. (ii) The implication of this property is that the constant component of the residual is resolved by the quadratic finite element space. The error, therefore, depends on the derivative of the residual. (iii) Since the error for quadratic solutions should be smaller than that for linear, the element Green’s function for linears should give an upper bound for the error of quadratics.
Application of variational a-posteriori multiscale error estimation to higher-order elements
3.1 Error upper bounds
3.3 Error estimates in the H 1 seminorm
Substituting (5) into (4) and applying the Hölder’s inequality [2], the following error upper bound can be derived
Taking the derivative of (4) with respect to x, corresponding upper bounds and exact error estimates can be calculated for the derivative error. Regarding the upper bound, if the norms exist, the same procedure as above yields
|u (x)| ≤ ||ge(2) (x, y)|| L p (ey ) ||Lu¯ − f || L q (e )
(7)
where 1 ≤ p, q ≤ ∞, 1/ p + 1/q = 1. Now, applying the L r norm to |u (x)|,
(2) |u ,x (x)| ≤ ||ge,x (x, y)|| L p (ey ) ||Lu¯ − f || L q (e ) ,
||u (x)|| L r (e ) ≤ ||ge(2) (x, y)|| L r (ex )×L p (ey ) ×||Lu¯ − f || L q (e ) .
(8)
where 1 ≤ p, q ≤ ∞, 1/ p + 1/q = 1. Application of a norm gives
(9)
(2) (x, y)|| L r (ex )×L p (ey ) ||u ,x (x)|| L r (e ) ≤ ||ge,x
In particular for p = q = r = 2,
upper
||u (x)|| L 2 (e ) ≤ τ L 2
||Lu¯ − f || L 2 (e ) ,
×||Lu¯ − f || L q (e ) .
where the intrinsic time-scale has been defined as upper
τL 2
= ||ge(2) (x, y)|| L 2 (ex ×ey ) .
(10)
For p = r = 1, q = ∞
||u (x)|| L 1 (e ) ≤ upper
=
(16)
For the exact error, one would arrive at (2)
upper meas(e )τ L 1 ||Lu¯
− f || L ∞ (e )
(11)
and τL 1
(15)
1 ||g (2) (x, y)|| L 1 (ex ×ey ) . meas(e ) e
u ,x (x) = −∇(Lu¯ − f )(0) b1,x (x) on e and (2)
(12)
|||u ,x (x)|||e = |∇(Lu¯ − f )(0)| |||b1,x (x)|||e
(17)
so the intrinsic parameter stems from the norm of the derivative of the residual-free bubble b1 (x).
3.2 Exact representation of the error for piecewise linear residuals In this section it is assumed that the residual is a piecewise linear function, i.e. Lu¯ − f ∈ P1 . This corresponds to the important case of advection-diffusion transport resolved with quadratic finite element spaces and f ∈ P1 . Then, the residual can be exactly expanded in Taylor series as [12]
4 Generalization to higher-order elements Likewise, for the class of problems considered in this article, the results of previous sections can be generalized as follows. Let k denote the polynomial degree of the finite element solution space.
(Lu¯ − f )(y) = (Lu¯ − f )(0) + ∇(Lu¯ − f )(0) y. Substitution in (4) and using the symmetry property (6) yields u (x) = − ge(2) (x, y) e
× (Lu¯ − f )(0) + ∇(Lu¯ − f )(0) y d y , = −∇(Lu¯ − f )(0) ge(2) (x, y) y d y , = −∇(Lu¯ −
e f )(0) b1(2) (x)
on e ,
≤ ||ge(k) (x, y)|| L r (ex )×L p (ey ) ||Lu¯ − f || L q (e )
(18)
or (13)
(2)
(2)
Applying the L r norm to the upper bound of |u (x)|, one arrives at ||u (x)|| L r (e )
where b1 (x) is the first order residual-free bubble [3–5, 10], that is, the asymmetric first moment of the element (2) Green’s function ge . Typically, the coordinate y is taken at the element center. See also [20] for a relation of residualfree bubbles with a posteriori ever estimation. Now, applying a norm to the above equality, the following exact error estimate is attained, |||u (x)|||e = |∇(Lu¯ − f )(0)| |||b1 (x)|||e .
4.1 Upper bound
(14)
Remark Note that the error for quadratic elements depends on the derivative of the residual.
||u (x)|| L r (e ) ≤ meas(e )s τ L r
upper
||Lu¯ − f || L q (e )
(19)
with upper
τLr
=
1 ||g (k) (x, y)|| L r (ex )×L p (ey ) meas(e )s e
(20)
and s = 1/r − 1/q.
4.2 Exact error for residuals in Pk−1 Due to the orthogonality property of the element Green’s function with respect to polynomials of order k − 2 [19]
Guillermo Hauke et al.
(a property that at least applies to one-dimensional H01 optimal solutions), the error (4) can be represented as y k−1 k−1 u (x) = − ge(k) (x, y) ¯ f )(0) d y , ∇ (Lu− (k−1)
0.2 0.15 0.1
b1 (2)|a|/(he )2
ey
1 k−1 ¯ f )(0) ge(k) (x, y) y k−1 d y , ∇ (Lu− =− (k−1) ey
0.05 0 -0.05
1 (k) ∇ k−1 (Lu¯ − f )(0)bk−1 (x) on e (21) =− (k − 1)!
-0.1 0
0.2
(k) bk−1 (x)
with the k − 1th order residual-free bubble for the polynomial basis of order k, i.e (k) bk−1 (x) = ge(k) (x, y) y k−1 d y on e . (22)
0.6
0.8
1
x/he (2)
Fig. 1 Nondimensional b1 (x) function for the hyperbolic case
x2 xh e − . (27) 2a 3a Note that, in the hyperbolic case, the homogeneous boundary conditions can only be applied to the inflow side of the element. Then, the upper bound intrinsic scales for error estimation are given as follows: he upper τL 2 = , (28) 3|a| he 1 1 1 upper − ln , (29) τL 1 = |a| 64 2 2 (2)
b1 (x) =
ey
Finally, taking a norm to both sides |||u (x)|||e =
0.4
|∇ k−1 (Lu¯ − f )(0)| (k) |||bk−1 (x)|||e (k − 1)!
(23)
Remarks (i) In general, a polynomial basis of order k > 1 resolves the residual up to the derivative of order k − 2, being the error dependent upon the k − 1 derivative of the residual. (ii) Therefore, the exact error can be represented as the product of the k − 1 residual derivative and the residual-free bubble of order k − 1. (iii) For elements of order k, a first definition of the flow time-scale parameter could be taken as the L 1 norm of the element Green’s function for the polynomial orde k.
5 The one-dimensional convection-diffusion equation
2 (2) , (30) = ||g,x (x, y)|| L 2 (e ×e ) = √ 3|a| where the last one estimates the upper bound for the derivative. And the residual-free bubble norms for exact error estimation yield upper
νL 2
(2)
The differential operator that defines the examples of this section is Lu = au ,x − κu ,x x ,
(24)
where a is the velocity and κ a diffusivity coefficient.
τ Lexact = 2
||b1 (x)|| L 2 (e ) he , = √ √ he he 3 30|a|
(31)
(2)
||b1 (x)|| L 1 (e )
4h e . 81|a| he The scale to estimate the error in the derivative is
τ Lexact = 1
2
=
(32)
(2)
||b1,x (x)|| L 2 (e ) 1 = = . √ 3|a| he he
5.1 Hyperbolic limit
ν Lexact 2
The element Green’s function for linears is 1 x > y, (1) ge (x, y) = a 0 x < y.
Remark It is very important to note that in the hyperbolic limit, the residual-free bubble is not zero at the element center. This suggests that the finite element solution is not exact at the interior nodes.
(25)
According to [19], that for quadratic elements can be calculated as 2 x he − y ge(2) (x, y) = ge(1) (x, y) − (26) a he he so the first order residual-free bubble is (see Fig. 1)
(33)
5.2 Diffusive limit For this limiting case, the element Green’s function for linears is
Application of variational a-posteriori multiscale error estimation to higher-order elements
0.02
5.3 Convection-diffusion
0.015
For the advection-diffusion equation, the element Green’s function for linear elements can be expressed as [17] ⎧ −ay/κ −ah /κ(1−x/ h ) e e 1−e 1−e ⎪ ⎪ x > y, ⎨ a (1−e−ah e /κ ) (1) ge (x, y) = (43) −ax/κ e−ah e /κ −e−ay/κ ⎪ ⎪ ⎩ 1−e x < y. a (1−e−ah e /κ )
b1(2)κ/(he)3
0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 0
0.2
0.4
0.6
0.8
1 0.2
x/he (2)
Fig. 2 Nondimensional b1 (x) function for the diffusive case
ge(1) (x, y) =
1 κh e (h e − x)y 1 κh e x(h e − y)
x>y x
