Name

gf_compute — Various computations involving the solution U of the finite element problem.

Calling Sequence

N = gf_compute(mf, U, 'L2 norm', mesh_im MIM [,ivec CVLST])
N = gf_compute(mf, U, 'H1 semi norm', mesh_im MIM [,CVLST])
N = gf_compute(mf, U, 'H1 norm', mesh_im MIM [,ivec CVLST])
N = gf_compute(mf, U, 'H2 semi norm', mesh_im MIM [,CVLST])
N = gf_compute(mf, U, 'H2 norm', mesh_im MIM [,ivec CVLST])
DU = gf_compute(mf, U, 'gradient', mesh_fem mfgrad)
D2U = gf_compute(mf, U, 'hessian', mesh_fem mfhess)
U2 = gf_compute(mf, U, 'interpolate on', mesh_fem mf2)
U2 = gf_compute(mf, U, 'interpolate on', mesh_slice sl)
[U2[,mf2,[,X[,Y[,Z]]]]] = gf_compute(mf,U,'interpolate on Q1 grid', 
      {'regular h', hxyz | 'regular N',Nxyz | X[,Y[,Z]]})
U2 = gf_compute(mf, U, 'extrapolate on', mesh_fem mf2)
E = gf_compute(mf, U, 'error estimate',  mesh_im MIM) 
gf_compute(mf, U, 'convect', mesh_fem mf_v, vec V, scalar dt, int nt[, string option]) 
  

Description

The first two arguments of this function are always mf and U, where U is a field defined on the mesh_fem mf.

  • n = gf_compute(MF, U,('L2 norm',MeshIm mim[, mat CVids]) Compute the L2 norm of the (real or complex) field U. If CVids is given, the norm will be computed only on the listed convexes.

  • n = gf_compute(MF, U,('H1 semi norm',MeshIm mim[, mat CVids]) Compute the L2 norm of grad(U). If CVids is given, the norm will be computed only on the listed convexes.

  • n = gf_compute(MF, U,('H1 norm',MeshIm mim[, mat CVids]) Compute the H1 norm of U. If CVids is given, the norm will be computed only on the listed convexes.

  • n = gf_compute(MF, U,('H2 semi norm',MeshIm mim[, mat CVids]) Compute the L2 norm of D^2(U). If CVids is given, the norm will be computed only on the listed convexes.

  • n = gf_compute(MF, U,('H2 norm',MeshIm mim[, mat CVids]) Compute the H2 norm of U. If CVids is given, the norm will be computed only on the listed convexes.

  • DU = gf_compute(MF, U,('gradient',MeshFem mf_du) Compute the gradient of the field U defined on MeshFem mf_du. The gradient is interpolated on the MeshFem mf_du, and returned in DU. For example, if U is defined on a P2 MeshFem, DU should be evaluated on a P1-discontinuous MeshFem. mf and mf_du should share the same mesh. U may have any number of dimensions (i.e. this function is not restricted to the gradient of scalar fields, but may also be used for tensor fields). However the last dimension of U has to be equal to the number of dof of mf. For example, if U is a [3x3xNmf] array (where Nmf is the number of dof of mf), DU will be a [Nx3x3[xQ]xNmf_du] array, where N is the dimension of the mesh, Nmf_du is the number of dof of mf_du, and the optional Q dimension is inserted if Qdim_mf != Qdim_mf_du, where Qdim_mf is the Qdim of mf and Qdim_mf_du is the Qdim of mf_du.

  • HU = gf_compute(MF, U,('hessian',MeshFem mf_h) Compute the hessian of the field U defined on MeshFem mf_h. See also gf_compute(MF, U,('gradient', MeshFem mf_du).

  • Ui = gf_compute(MF, U,('interpolate_on',list(MeshFem mfi | Slice sli)) Interpolate a field on another MeshFem or a Slice.

    • Interpolation on another MeshFem mfi: mfi has to be Lagrangian. If mf and mfi share the same mesh object, the interpolation will be much faster.

    • Interpolation on a Slice sli: this is similar to interpolation on a refined P1-discontinuous mesh, but it is much faster.

    This can also be used with gf_slice('points') to obtain field values at a given set of points. See also gf_asm('interpolation_matrix')

  • Ue = gf_compute(MF, U,('extrapolate_on',MeshFem mfe) Extrapolate a field on another MeshFem. If the mesh of mfe is stricly included in the mesh of mf, this function does stricly the same job as gf_compute(MF, U,('interpolate_on'). However, if the mesh of mfe is not exactly included in mf (imagine interpolation between a curved refined mesh and a coarse mesh), then values which are slightly outside mf will be extrapolated. See also gf_asm('extrapolation_matrix')

  • E = gf_compute(MF, U,('error_estimate',MeshIm mim) Compute an a posteriori error estimation. Currently there is only one which is available:

    for each convex, the jump of the normal derivative is integrated on its faces.

  • [U2[,MF2,[,X[,Y[,Z]]]]] = gf_compute(MF,U,'interpolate on Q1 grid',{'regular h', hxyz | 'regular N',Nxyz |X[,Y[,Z]]} Creates a cartesian Q1 mesh fem and interpolates U on it. The returned field U2 is organized in a matrix such that in can be drawn via the MATLAB command 'pcolor'. The first dimension is the Qdim of MF (i.e. 1 if U is a scalar field). Example (mf_u is a 2D mesh_fem):

      
    Uq = gf_compute(mf_u, U, 'interpolate on Q1 grid', 'regular h', [.05, .05]);
    pcolor(squeeze(Uq(1,:,:)));
     
  • E = gf_compute(MF, U, 'mesh edges deformation', N [,vec or mat CVLIST]) [OBSOLETE FUNCTION! will be removed in a future release]

    Evaluates the deformation of the mesh caused by the field U (for a 2D mesh, U must be a [2 x nb_dof] matrix). N is the refinment level (N>=2) of the edges. CVLIST can be used to restrict the computation to the edges of the listed convexes (if CVLIST is a row vector), or to restrict the computations to certain faces of certain convexes when CVLIST is a two-rows matrix, the first row containing convex numbers and the second face numbers.

  • UP = gf_compute(MF, U, 'eval on triangulated surface', int Nrefine,[vec CVLIST]) [OBSOLETE FUNCTION! will be removed in a future release]

    Utility function designed for 2D triangular meshes: returns a list of triangles coordinates with interpolated U values. This can be used for the accurate visualization of data defined on a discontinous high order element.

    On output, the six first rows of UP contains the triangle coordinates, and the others rows contain the interpolated values of U (one for each triangle vertex) CVLIST may indicate the list of convex number that should be consider, if not used then all the mesh convexes will be used. U should be a row vector.

  • gf_compute(mf, U, 'convect', mesh_fem mf_v, vec V, scalar dt, int nt, string option]) Compute a convection of U with regards to a steady state velocity field V with a Characteristic-Galerkin method. This method is restricted to pure Lagrange fems for U. mf_v should represent a continuous finite element method. dt is the integration time and nt is the number of integration step on the caracteristics. option is an option for the part of the boundary where there is a re-entrant convection. option = 'extrapolation' for an extrapolation on the nearest element or option = 'unchanged' for a constant value on that boundary. This method is rather dissipative, but stable.

Examples

Using the error estimate to refine the worst convexes:

  
E = gf_compute(mf, U, 'error_estimate', mim);
gf_mesh_set(m, 'refine', find(E < 1e-3));  
 

See Also

gf_mesh

Authors

Y. Collette