27 where \f$\boldsymbol{v},\phi \in \boldsymbol{V}\f$, \f$p,q \in \boldsymbol{W}\f$ and \f$\boldsymbol{V},\boldsymbol{W}\f$ are appropriate spaces for the velocity
28 and the pressure system to satisfy the inf-sup condition.
29 Using a matrix notation the discrete system may be written as:
30 \f{displaymath}
31 \left[ \begin{array}{ccc}
32 A & D_b^T & B\\
33 D_b & 0 & D_i^T\\
34 B^T & D_i & C
35 \end{array}\right]
36 \left[ \begin{array}{c}
37 \boldsymbol{v_b}\\
38 p\\
39 \boldsymbol{v_i}
40 \end{array}\right] =
41 \left[ \begin{array}{c}
42 \boldsymbol{f_b}\\
43 0\\
44 \boldsymbol{f_i}
45 \end{array}\right]
46 \f}
47
48 where the components of \f$A,B\f$ and \f$C\f$ are \f$(\nabla\phi_b,\nu\nabla\boldsymbol{v_b})\f$, \f$(\nabla\phi_b,\nu\nabla\boldsymbol{v_i})\f$ and
49 \f$(\nabla\phi_i,\nu\nabla\boldsymbol{v_i})\f$ and the components \f$D_b\f$ and \f$D_i\f$ are \f$(q,\nabla\boldsymbol{v_b})\f$ and \f$(q,\nabla\boldsymbol{v_i})\f$.
50 The indices \f$b\f$ and \f$i\f$ refer to the degrees of freedom on the elemental boundary and interior, respectively. In constructing the system we have lumped the
51 contributions form each component of the velocity field into matrices \f$A,B\f$ and \f$C\f$. However we note that for a Newtonian fluid the contribution from
52 each field is decoupled. Since the inetrior degrees of freedom of the velocity field do not overlap, the matrix \f$C\f$ is block diagonal and to take advantage
53 of this structure we can statically condense out the \f$C\f$ matrix to obtain the system:
54 \f{displaymath}
55 \left[ \begin{array}{ccc}
56 A-BC^{-1}B^T & D_b^T-BC^{-1}D_i & 0\\
57 D_b-D_i^TC^{-1}B^T & -D_i^TC^{-1}D_i & 0\\
58 B^T & D_i & C
59 \end{array}\right]
60 \left[ \begin{array}{c}
61 \boldsymbol{v_b}\\
62 p\\
63 \boldsymbol{v_i}
64 \end{array}\right] =
65 \left[ \begin{array}{c}
66 \boldsymbol{f_b} - BC^{-1}\boldsymbol{f_i}\\
67 -D_i^TC^{-1}\boldsymbol{f_i}\\
68 \boldsymbol{f_i}
69 \end{array}\right]
70 \f}
71
72 To extend the aboce Stokes solver to an unsteady Navier-Stokes solver we first introduce the unsteady term, \f$\boldsymbol{v_t}\f$, into the Stokes problem.
73 This has the principal effect of modifying the weak Laplacian operator \f$(\nabla\phi,\nu\nabla\boldsymbol{v})\f$ into a weak Helmholts operator
74 \f$(\nabla\phi,\nu\nabla\boldsymbol{v})-\lambda(\phi,\boldsymbol{v})\f$ where \f$\lambda\f$ depends on the time integration scheme. The second modification requires
75 the explicit discretisation of the non-linear terms in a similar manner to the splitting scheme and this term is then introduced as the forcing term \f$\boldsymbol{f}\f$.
76
77\section sectionRunning1example_cuopled Running a first example
78
79 The folder <code>Nektar++/regressionTests/Solvers/IncNavierStokesSolver/InputFiles</code> contains several <code>*.xml</code> files.
80 These are input files for the Navier-Stokes solver specifying the geometry (i.e. the mesh and
81 the spectal/hp expansion), the parameters and boundary conditions. Further details on the structure
82 of the input file can be found in @ref pageXML.
83
84 Now, lets try to run one of them with the coupled solver.
85
86 - Copy the input file, <code>Test_ChanFlow_LinNS_m8.xml</code> to the directory where the solver is compiled, e.g.
157 \image html Oseen.png "Steady Oseen Flow (u-velocity component)"
158
159
160\section sectionRef_cuopled References
161 [1] S.J. Sherwin and M. Ainsworth: <i>Unsteady Navier-Stokes Solvers Using Hybrid Spectral/hp Element Methods</i>, Conference Paper, 2000.<br />
162 [2] R. Stenberg and M. Suri: <i>Mixed hp finite element methods for problems in elasticity and Stokes flows</i>, Numer. Math, 72, 367-389, 1996.<br />
163 [3] M. Ainsworth and S.J. Sherwin: <i>Domain decomposition preconditioners for p and hp finite element approximation of the Stokes equations</i>, Comput. Methods Appl. Mech. Engrg., 175, 243-266, 1999. <br />
164 [4] P. La Tallec and A. Patra: <i>Non-overlapping domain decomposition methods for adaptive hp approximations of the Stokes problem with discontiunous pressure field</i>, Comput. Methods Appl. Mech. Engrg., 145, 361-379, 1997.<br />
165 [5] G. E. Karniadakis, M. Israeli, and S. A. Orszag: <i>High-order splitting methodsfor the incompressible Navier-Stokes equations</i>, J. Comput. Phys., 97, 414-443, 1991.<br />