Comparison of SUPG with Bubble Stabilization Parameters and the Standard SUPG

and Applied Analysis 3 Obviously the sumof first three terms is greater thanA|||V h ||| 2 from the condition of (2), and we only need to estimate the last term. Using Hölder inequality, we have 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∑

When the mesh Péclet number Pe is greater than 1, there exist global unphysical oscillations in numerical solutions of standard discretization schemes on general meshes.Hence, stabilized methods and/or a priori adapted meshes are widely used in order to get discrete solutions with satisfactory accuracy.An overview on these methods can be found in the survey [1,2].
One of the most famous stabilized finite element methods is the streamline upwind Petrov-Galerkin method (SUPG).The SUPG proposed by Hughes and Brooks [3] is known to provide good stability properties and high accuracy.However, there are several drawbacks in SUPG, such as lacking discrete maximum principle and involving second derivatives and difficulties in determining the stabilization coefficients.Driven by these problems, many researchers were devoted to improving the SUPG.A lot of numerical methods were proposed, such as SOLD [4], nonlinear residual [5], and LPS [6].Also, relations of SUPG and other numerical formulation, like residual-free bubble method [7] and variational multiscale method [8], were studied to seek possible directions of improvement.
By means of residual, the SUPG adds to the original bilinear form a term which introduces a suitable amount of artificial viscosity in the direction of streamlines.Also, the SUPG can be viewed as an inconsistent Petrov-Galerkin 2 Abstract and Applied Analysis method, since its modified weighting function cannot apply to the diffusive term (see the details in Section 4).
For this reason, we are to analyze the SUPG with bubble stabilization coefficients in 2D and compare its numerical performance with the SUPG's.From theoretical analysis and numerical results, we find that the new scheme is classified into the consistent Petrov-Galerkin formulation (CPGF) and behaves as well as SUPG.Also, the standard finite element method (FEM), which shows excellent performances for Pe ≤ 1, can be classified into the CPGF and viewed as a special "SUPG with bubble stabilization coefficients" by taking   = 0 in (7).Thus, the FEM and our new scheme could be viewed as two reference numerical methods in the CPGF.This provides possibilities of constructing new numerical schemes between them in the CPGF.In fact, in our forthcoming works, we obtain a linear maximum-principle-preserving stabilized method in the CPGF by means of the FEM and the SUPG with bubble stabilization coefficients, which shows better numerical performances than the standard SUPG.Thus, our results in this paper can be viewed as a starting point to construct new numerical schemes in the CPGF.
The remainder of this paper is organized as follows.In Section 2 we formulate the problem and introduce notation and mesh.Theoretical results including stability analysis and energy norm estimates can be found in Section 3. Section 4 is devoted to the relationship between SUPG, standard finite element method, and our method.Finally, numerical experiments that illustrate our theoretical results are presented in Section 5.

Mesh and Numerical Formulation
First we define a finite element space on triangular meshes where the term "linear" is to be understood in the usual isoparametric sense.Here we assume that the triangulations T ℎ on Ω are quasiuniform: for any  ∈ T ℎ , where  0 is a positive constant and ℎ  ,   , ℎ denote, respectively, the diameter of , the smallest angle of , and the maximum of all diameters of triangles in T ℎ .
Using the linear finite element space  ℎ , we can state the standard Galerkin discretisation of (1) which reads as follows.
The SUPG consists in adding to the original bilinear form a term which introduces a suitable amount of artificial viscosity in the direction of streamlines.In this case, the SUPG reads as follows.
Finally, we define a special energy norm (SD norm) associated with  ℎ (⋅, ⋅): We denote by ‖ ⋅‖  the  2 norm in  2 (); that is, If  = Ω, then we drop Ω from the notation.

Stability and Energy Estimates
Throughout this subsection, we assume  is some positive constant.

Stability Analysis.
The stability properties are a consequence of the following.
Proof.By divergence theorem we obtain Abstract and Applied Analysis 3 Obviously the sum of first three terms is greater than |||V ℎ ||| 2 from the condition of ( 2), and we only need to estimate the last term.Using Hölder inequality, we have where ( 14) is based on the definition of   , (8), and ( 11): Then the proof of the lemma is finished.

Energy Norm Estimate.
We denote by   the nodal piecewise linear interpolant to  over T ℎ .From Lemma 1 and the fact of  ℎ ( −  ℎ ,   −  ℎ ) = 0 one gets Denote  :=   − ,  :=   −  ℎ and estimate the right-hand side of (16) term by term: where we have used the standard interpolation results ‖  ‖≤ ℎ 2 (see [9]) and the first inequality of ( 18) is obtained by Combining all of these estimates, one gets

Comparison of SUPG with Bubble Stabilization Coefficients and the Standard SUPG
Consider the bilinear form of SUPG: where   is constant in .
It can be rewritten in the form where δ|  =   .
Notice that (22) does not correspond to a consistent Petrov-Galerkin formulation in general except in the case when b ≡ C and V ℎ is linear.Clearly, when   is a bubble function, since   vanishes on the boundary of .Thus (22) can be written as Then SUPG with bubble stabilization coefficients is classified into the consistent Petrov-Galerkin formulation.Moreover, in general, piecewise constants   in SUPG make the test function V ℎ +  b⋅∇V ℎ discontinuous.However, the test functions are continuous in the case of bubble stabilization coefficients and the consequent test space  ℎ is contained in  1 0 (Ω).In this case, SUPG with bubble stabilization coefficients reads as follows.
Find  ℎ ∈  ℎ , such that for all  ℎ ∈  ℎ , where On the other hand, SUPG with bubble stabilization coefficients is gradually close to the FEM in the same space  1 0 (Ω) as  *  → 0 for any ∈ T ℎ .In a word, SUPG with bubble stabilization coefficients inherits the advantages of SUPG and constructs the relation between FEM and SUPG in consistent Petrov-Galerkin formulations.

Numerical Experiments
In this section we give numerical results that appear to support our theoretical results.Errors and convergence rates for our numerical scheme are presented.All calculations are carried out by using Intel visual Fortran 11.The discrete problems are solved by using a version of Pardiso solver (see [10,11]).
For the computations we have chosen   = 60.0ℎ 1  2  3 in SUPG with bubble stabilization coefficients and   = 1.0ℎ in SUPG.We set Ω = [0, 1] 2 and calculate the errors and convergence rates in the subdomain Ω  away from layers for Tables 1-16 where  is chosen such that the solution  is where  is chosen such that the solution  is Problem 4. One has From the above tables it is shown that the errors and convergence rates of SUPG with bubble stabilization coefficients and standard SUPG in the maximum norm, in the  2 norm, and in the SD norm are similar not only in the subdomain away from layers but also in the global sense.These results illustrate that SUPG with bubble stabilization coefficients also has good stability properties and high accuracy as standard SUPG.