Star-based a Posteriori Error Estimator for Convection Diffusion Problems

In this paper, we derive an a posteriori error estimator, for nonconforming finite element approximation of convection-diffusion equation. The a posteriori error estimator is based on the local problems on stars. Finally, we prove the reliability and the efficiency of the estimator without saturation assumption nor comparison with residual estimator


Introduction
A posteriori error estimators provide the basis for adaptive mesh refinement and quantitative error control [1,8,4,5,12,19,14,10]. One of the most successful estimators was proposed by Bank and Weiser and extended by many authors [2,3,7,13,16,20,21], it is based on the solution of local Neumann problems on elements, which seem to allow for cancelation and thus lead to better results than the residual estimators. The classical proof of equivalence with the energy error require the saturation assumption : this says that the solution can be approximated asymptotically better with quadratic than with linear finite elements. The saturation assumption is shown to be superfluous by Nochetto in [17]. However, removing this assumption requires comparison with residual estimators. More recently, a new a posteriori error estimators on stars was proposed in [15], and the proof of the equivalence with energy error it applies directly without reference to residual estimators.
In this paper, we extended the results of [15] to the case of nonconforming finite elements and the convection diffusion case. A new a posteriori error estimator is introduced based on the solution of a small discrete problem in stars. We prove the reliability and the efficiency of the estimator without saturation assumption nor comparison with residual estimator. We consider the simpler case of nonconforming approximations for convection diffusion problem, and we introduced a technique which allowed us to define a new a posteriori error estimator which are equivalent to the energy error.

Setting the problem
We consider here the convection-diffusion problem: (P )    −ε∆u + β.∇u = f in Ω, u = 0 on Γ = ∂Ω In the following we assume that Ω ∈ R 2 a simply connected polygon domain, 0 < ε << 1 , β ∈ (W 1,∞ (Ω)) 2 ,, such that − 1 2 div β ≥ a > 0 and f ∈ L 2 (Ω). Let T h be a family of conforming shape-regular triangulation of Ω by triangular, we denoted by E I the set of interior edges and by E f the set of all edges included in Γ. Let V h be the Universal Journal of Applied Mathematics 2(1): 50-62, 2014 51 lowest order non-conforming Crouzeix-Raviart finite element space defined by: where [.] E denoted the jump of the function across E. For each T ∈ T h ,we denote by P k (T ) the polynomial space of degree less than or equal to k.
For all T ∈ T h , we define ∂T − such the part of the frontier of T such that β.n T < 0 where n T stands for the unit outward normal vector to T on ∂T .
In the sequel, we consider u N C h ∈ V h be a solution of the stabilized nonconforming approximation problem:

The a posteriori error estimator
For the a posteriori error analysis of the considered approximation, we need to define some local spaces and problems. We denoted by {x i } i∈N the set of all nodes of the triangulation T h . For each i ∈ N , ϕ i denoted the canonical continuous piecewise linear basis function corresponding to x i . The star ω i is the interior relative to Ω of the support of ϕ i , and h i is the maximal size of the elements forming ω i . Finally, Γ i will denote the union of the sides touching x i that are contained in Ω, and Γ i will denote the union of the sides touching x i that are contained in Ω.
if x i is an interior node, and We have the following result of ([15]) Proposition 1 There exists a constant C, only depending on the minimum angle of the triangulation but independent of the star being considered, such that: We define the finite dimensional local spaces P 2 0 (ω i ) as follows,

Definition 2
For i ∈ N , let P 2 (ω i ) denote the space of continuous piecewise quadratic functions on star ω i that vanish on ∂ω i . The spaces In the following we consider the energy norm: Let u N C h ∈ V h be fixed and we denoted by ∇ h u h the vector belonging to (L 2 (Ω)) 2 defined by

52
Star-based a Posteriori Error Estimator for Convection Diffusion Problems For each i ∈ N , we consider the local problems : Using Lax-Milgram theorem, we prove that the discrete problems have unique solution. The problems (P 1 ) i estimate the approximation error, but the problems (P 2 ) i estimate the consistency error of the used method.
Finally we set:

Upper bound of the error
In this section we prove the one of main results of this paper. First, we prove the upper bound of the error without oscillation. As in ( [6], [9], [11]). Recall that [18]:

Lemma 3 (Discrete Poincar and Friedrichs inequalities ).
There is a positives constants C depending only on the minimum angle of T h the Ω such that: We have the following global upper bond of the error: There is a positive constant C 1 depending only on the minimum angle of T h such that ( where osc(f ) is the data oscillations defined by: Proof. Remark that using Helmholtz-decomposition, we have with w ∈ H 1 0 (Ω), ζ ∈ H 1 (Ω) and ∫ Ω ∇w.Curl ζdx = 0.
Let us remark also that the orthogonality implies the following error decomposition : ,Ω + ε∥Curl ζ∥ 2 0,Ω . and the following equalities : and The estimates of expressions in (5) and (6) will be established respectively in the Lemmas 7 and 8 . As a main tool we use the following Lemmas.

Lemma 5 For each node i ∈ N there exists an unique operator
, such that for any v ∈ V (ω i ) the following conditions hold : where the constant C depends only on the minimum angles of T h .
The Lemma 5, is an adaptation of arguments given in [15], and so the proof will be skipped. and To prove the second equality, if we denoted by [ ∂u N C h ∂τE ] ∈ P 0 (E) the jump of the tangential derivative across the edge E. We have by Green formula and lemma 5 ∫ The following Lemma give an estimate of expression (5): Lemma 7 For w ∈ H 1 0 (Ω) defined in (4), there exists positives constants C * , depending only on the minimum angle of T h such that Proof . For w ∈ H 1 0 (Ω) defined in (4), we set for interior nodes, and w i = 0 otherwise.
By adapting standard arguments used in the analysis of finite element approximation, and by using the propriety − 1 2 div β ≥ a > 0, wa have: This gives, Since w h ∈ V h ∩ H 1 0 (Ω), a(w, w h ) = 0, we have: a(w, w) = a(w, w − w h ). Then Universal Journal of Applied Mathematics 2(1): 50-62, 2014 55 Since (w − w i ) ∈ V (ω i ), adding and removing same quantities in the two last terms give: Using the definition of local problems (P 1 ) i , We now process successively with each term of the right-hand side. On one hand, using Cauchy-Schwarz and item 3 of lemma 5 we have: On the other hand, since both of (w −w i ) and Π i (w −w i ) belong to V (w i ), using definition of V (w i ) and coefficients f i give: Star-based a Posteriori Error Estimator for Convection Diffusion Problems Using Cauchy-Schwarz, the proposition 1, the item 4 of lemma 5 and once more C is a generic constant only depending on the minimum angle of triangulation. We note: Using the Green formula we have: Using Cauchy-Schwarz and the proposition 1 we have: Summing up the different contributions in the estimate of ∥w∥ ε,ωi and using the continuity of a(., .) yield the result. We have also the following result giving an estimation of expression (6): (4), there exists a positive constant C, depending only on the minimum angle of T h such that Proof. Let ζ ∈ H 1 (Ω) defined in (4), we note for all nodes.
Which easy to verify that: and hence: On one hand, since: Curl ζ h ∈ H(div ; Ω), div(Curl ζ h ) = 0, and ∀T ∈ T h , (Curl ζ h ) |T belonging the lowest Raviart-Thomas space RT 0 (T ) = (P 0 (T )) 2 + xP 0 (T ) and On the other hand, since (ζ − ζ i ) ∈ V (ω i ), using Lemma 5, and the definition of local problem (P 2 ) i we have, for Using the last equalities, ∑ i∈N ϕ i = 1 and the equality ∥Curl ζ∥ 0,Ω = ∥∇ζ∥0, Ω , we obtain Which completes proof of the lemma. The combination of the two lemmas 7 and 8, gives the result of the upper bound of the error of the theorem 4.

Lower bound of the error
In this section we prove a lower bound of the error without oscillation.

Theorem 9
Let u N C h ∈ V h , there exists a positives constants C 1 and C 2 , depending on the minimum angle of the triangulation such that, for any i ∈ N , Proof. Let i ∈ N . We have Then Using Cauchy-Schwarz theorem imply that: Or: Then ε And: ∑ Summing up the different contributions in the estimate of E 1,i (u N C h ) we get: To prove the second inequality, let us remark that, using Green formula, we have : Then using Cauchy-Schwartz inequality we deduce Where ε Then with u = 0 on ∂Ω such that β = [3, 2] t , ϵ = 0.01, and f = 1 This solution, for small ϵ > 0, exhibits boundary layers near the top (y = 1) and right (x = 1) boundaries. The adaptive finite element method correctly refines in these layers, yielding accurate solutions with a small number of unknowns (relative to uniform refinement). The meshes and the contour maps are omitted for brevity reasons (see Fig 1 and Fig 2).
The interesting point of this problem concerns the exactness of estimators as ϵ goes to zero   which presents sharp curvature in the vicinity of point (0.5, 0.5), and we perform a nonconforming finite element discretization on it. Successive iterations of adaptive mesh are represented in Figure 4. Computed and Exact solution are given in Figure 5, where the scaling of the height is the same for both pictures.

Remarks.
1. We can write a general framework regroups the conforming and nonconforming approximations of the convection diffusion problem, just write the approximate problem as: 0 (Ω)) in the conforming case and W h = V h in the non conforming case. The same arguments can be used to derive an a posteriori error estimate on stars with similar properties.
2. The three dimensional case can be made with some regularity assumptions of the domain Ω, and by adapting the used technics in the two dimensional case.

Conclusion
In this work we analyzed an a posteriori error estimator for nonconforming convection diffusion problem, with the Helmholtz-Decomposition technics. These estimators are efficient and robust to respect to physical parameters of the problem.