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# An Optimal Error Estimate for a Nonconforming Finite Element Method of Upwind Type Applied to the Stationary Navier–Stokes Equations in Two and Three Dimensions

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An Optimal Error Estimate for a Nonconforming Finite Element Method of Upwind Type Applied to the Stationary Navier–Stokes Equations in Two and Three Dimensions book

# An Optimal Error Estimate for a Nonconforming Finite Element Method of Upwind Type Applied to the Stationary Navier–Stokes Equations in Two and Three Dimensions

DOI link for An Optimal Error Estimate for a Nonconforming Finite Element Method of Upwind Type Applied to the Stationary Navier–Stokes Equations in Two and Three Dimensions

An Optimal Error Estimate for a Nonconforming Finite Element Method of Upwind Type Applied to the Stationary Navier–Stokes Equations in Two and Three Dimensions book

## ABSTRACT

For the finite element approximation we use the nonconforming PI /PO element pair of Crouzeix/Raviart [2] . Let Th be a regular decomposition of the domain 0 C Rd into d-dimensional simplices K E Th where the mesh parameter h represents the maximum diameter of the elements kth . We denote by ri the (d - I )-dimensional faces of the elements K E T h assuming inner faces for i = 1 , . . . , N and boundary faces ri C ao for i = N + 1 , . . . , N + M. Bi is supposed to be the barycentre of ri o Now we can define the discrete spaces Vh � V and Q h � Q by

where Pm(K) , m = 0 , 1 , is the set of all polynomials on K with degree not greater than m. Note that a function v E Vh in general is discontinuous on the faces ri of the elements K E Th which implies Vh ct. V. Therefore, we need for the discretization of the weak formulation (2) the elementwise defined bilinear forms

(7)

but we apply an upwind technique proposed in [5 ] , [7] . This technique is based on a decomposition of the domain n into so called lumping regions Ri which are assigned to the faces ri. Let OK be the barycentre of the element K and SK,l the d-dimensional simplex contained in K which has r i as one of its faces and the point OK as additional vertex. We denote by Ai the set of all indices k =I-£ for which the nodes Bk and Bi belong to a common element K and define in this case r ik as the common (d - 1 ) dimensional face of SK,i and SK, k ' For inner faces ri , there exist two elements K and K' with ri = aK n aK' (see Figure 1 ) and we define the lumping region Rl by Rl := SK,i U SKI , t . In the case of a boundary face rt c aK n an we set Rl : = SK,l . We define a lumping operator Lh , which transforms a given function v E Vh into a piecewise constant function LhV by

(8)

(9)

Possible choices for �O which have been already used in practical computations (see [6] , [4] , [ 13D are

respectively,wherewewillomittheindexwifw E n,andletI I.IIhbethefollowing discrete Hl -norm on V + Vh

( 14 ) By Wh we denote the subspace of discrete divergence-free functions

( 15 )

Now, we will recall from [7] , [ 10] , [ 11 ] two useful lemmata and two results concerning the existence of solutions and the convergence of our upwind discretization ( 1 1 ) in the case of an arbitrary Reynolds number Re = V-l .