If I want use function space of tensor?

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1

If I need the function space of vector, I can use VectorL2(mesh, order). But if I need the function space of tensor, what I should use?

Another question is that when we solve saddle point problem, like Stokes problem, the matrix is indefinite, how the solve it? what command can we use?

thanks

2

Hi yhhan,

there is currently no MatrixL2 space direktly available. However, you can “construct” it by

MatrixL2 = L2(mesh, order=order, dim=4)
u,v = MatrixL2.TnT()
u.dims = (2,2)
v.dims = (2,2)
gf = GridFunction(MatrixL2)
gf.dims = (2,2)

For a saddle point problem you can use direct solvers as umfpack or also the build-in sparsecholesky solver if you regularize the problem at the lower-right zero block. E.g. for Stokes involving velocity u and pressure p you can add

a += -1e-9*p*q

as regularization.

Best
Michael

3

a short note on the solver for saddle point problems:

you can easily use MinRes or GMRes with block-preconditioning, see e.g.
https://ngsolve.org/docu/latest/i-tutorials/unit-2.6-stokes/stokes.html
as a starting point.

Joachim

4

Hi Michael,

Thank you so much, I will try it.

best,

Yihui

5

Hi Joachim,

I will check it later, thank you for your help.

best,

Yihui

6

Hi Michael,

For vector space, I can use V=VectorL2(mesh, order = ), then in the following
MatrixL2 = L2(mesh, order=order, dim=4)
u,v = MatrixL2.TnT()
u.dims = (2,2)
v.dims = (2,2)
gf = GridFunction(MatrixL2)
gf.dims = (2,2)

which one is like the above space V written by me? It seems that MatrixL2 = L2(mesh, order=order, dim=4) is different from the space V.

Best,
Yihui

7

Hi Yihui,

writing

V = VectorL2(mesh, order=...)

gives you a vector and

V = L2(mesh, order=..., dim=4)

a 2x2 matrix (after reshaping with .dim=(2,2) for the corresponding Trial-, Test-, and GridFunctions).

Best
Michael