Abstract
We derive optimal reliability and efficiency of a posteriori error estimates for the steady Stokes problem, with a nonhomogeneous Dirichlet boundary condition, solved by a stable enriched Galerkin scheme (EG) of order one on triangular or quadrilateral meshes in ℝ2, and tetrahedral or hexahedral meshes in ℝ3.
Acknowledgment
Part of this work was developed while M.G. was visiting the Laboratoire Jacques-Louis Lions, at Sorbonne Université. M.G. would like to thank Ministerio de Ciencia, Innovación y Universidades (Spain) for its support through grant PRX19/00475; M.G. also acknowledges the support received from the Centro de Investigación de Galicia CITIC, funded by Xunta de Galicia and the European Union (European Regional Development Fund – Galicia 2014-2020 Program), by grant ED431G 2019/01 and to Xunta de Galicia (Spain) by its support through grant GRC ED431C 2018-033. The simulations presented in this paper were carried out in the supercomputer Finisterrae II; the authors would like to thank CESGA for letting them use its capabilities.
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A Approximation of the boundary data
In practical situations, the boundary data gD is given by measurements or by the results of a previous computation; in both cases, it is a discrete set of values. To simplify the a posteriori error analysis, it is convenient to assume that this discrete set is interpolated so as to be the trace of a function of
This is adequate since the function x is a polynomial of degree one, thus is in the discrete space, and its divergence is the constant d, the dimension; clearly,
Then the discrete set gD is approximated by
By construction, gh,D satisfies (2.1), and the discrete problem (3.5) is solved with the data gh,D. To simplify notation, the index h in gD was dropped in the previous analysis.
The same approach is used in the case of a manufactured solution, but in this case we can measure the error made when replacing gD with gh,D. For Πh, we use the Scott–Zhang interpolation operator of degree k in each element, globally continuous. First,
Now,
which is a bounded quantity and of course, ‖x‖[L2(Γ)]d is also a bounded quantity. Therefore the approximation error of gD is bounded as follows:
and
The a posteriori error analysis developed in the previous sections holds with gh,D instead of gD and (u, p) replaced by (u(h), p(h)) ∈ H1(Ω)d × L(Ω), solution of
As gh,D is a continuous piecewise polynomial, it belongs to [H1(Γ)]d and so there exists some s, 0 < s < 1/2, such that u(h) ∈ [H1+s(Ω)]d and p(h) ∈ Hs(Ω). Hence the error equalities of Section 4.1 are well-defined and all reliability and efficiency bounds derived above hold with u and p replaced by u(h) and p(h). To express the reliability bounds in terms of u and p, we use the following:
Regarding efficiency, we must add to the right-hand side of (4.51) the terms
and to that of (4.56),
These quantities can be bounded by the global estimates in (A.4), but this is not optimal. Sharper bounds can be derived by applying the localizing techniques of 34, but this is outside the scope of this work.
B Choice of penalty parameters
The penalty parameters σe are chosen to guarantee that the form aε is elliptic in
Thus,we must estimate
Therefore, when E1 and E2 are the two elements sharing e, we have
Now, we sum over all interior e:
Let us take the maximum of the factor
and observe that the second factor is of the order of h−1 , whatever the dimension. Therefore, e
where depends only on the regularity of the mesh. It remains to count the number of occurrences of an element E in the sum over e. Each face involves its two adjacent elements. Since a simplex has d + 1 faces, it is counted d + 1 times. Therefore,
In the case of a quadrilateral or hexahedral element, d + 1 is replaced by 2d, the number of faces, and the constantis multiplied by ̂D. Of course, this formula is also valid for boundary faces, except that the factor
C The case of hexahedra with curved faces
From the computer implementation point of view, handling curved faces is more difficult, in particular because the normal vector to each face ne is now a function of its position x. But when the family of meshes is regular, this function does not vary much and from a theoretical point of view, all the preceding work applies immediately to regular families of hexahedra with straight edges but non planar faces, with the possible exception of the inf-sup condition. To establish the inf-sup condition in this case, we proceed as in 10, namely use Fortin’s lemma. This amounts to constructing a suitable approximation operator
In 10, such an operator is constructed by correcting a Scott–Zhang approximation operator, say Πh ∈
where c(v) is a constant vector in each cell. It is shown in 10 that a sufficient condition for the equality in (C.1) is that c(v) satisfy on all faces γi of E with interior normal , nγ
In the case when γi is a plane face, (C.2) reads
The regularity of the element E implies that this system is solvable provided the maximum number of such faces per element is no more than three in 3D or two in 2D and none are opposite faces, in other words, these faces are part of a conical angle. The situation is the same when γi is a curved face and the mesh is regular, because the normal vector to a face varies little. In this case (C.3) is replaced by
Again, we assume that an element has no more than d faces with interior normals and none are opposite faces. Take the 3D case of exactly three such faces, say γ1, γ2, γ3. When written in matrix form, (C.4) reads
where b is the vector with ith component
and the ith line of the matrix N is
Of course, the fact that nγi are unit normals imply that N is uniformly bounded. Moreover, the regularity assumption on the family of triangulations implies that N is nonsingular and its inverse is uniformly bounded. The second part of (C.1) follows from this.
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