Find L2 projection of a given function in ngsolve

dong · June 18, 2020, 5:04am

I’m finding L2 projection of a given function f on a finite element space Q.

Is there any function in Ngsolve do this job as in Fenics?

I also tried to write a find to find the L2-projection using its definition, but I met an error when finding the inverse matrix.

Could you please tell me how to fix this? Please see the attached file.
Thank you so much.

Attachment: Projection1.ipynb

joachim · June 18, 2020, 6:00am

You have to do it following its definition. You are building some sub-block in a huge matrix, and the solver is telling you correctly that the big matrix is not invertible

Build and invert the matrix on a single space, then you can embed the single-space vectors in your product-space,

Joachim

dong · June 18, 2020, 6:00pm

Thank you so much. Could you please show me how to embed a single-space vectors in a product space?
I searched a tutorial from ngsolve documentation, but I couldn’t find an example.

joachim · June 18, 2020, 7:35pm

You can use an embedding matrix:

r = IntRange (a,b)
emb = Embedding(n, r)

This embeds a vector of length b-a into a vector of length n, starting at index a.

some use-cases you can find here:

github.com

NGSolve/modeltemplates/blob/master/templates/NavierStokes.py

from ngsolve import *
from .mt_global import *

__all__ = ["NavierStokes"]

class NavierStokes:
    
    def __init__(self, mesh, nu, inflow, outflow, wall, uin, timestep, order=2, volumeforce=None):

        self.nu = nu
        self.timestep = timestep
        self.uin = uin
        self.inflow = inflow
        self.outflow = outflow
        self.wall = wall
        
        V1 = HDiv(mesh, order=order, dirichlet=inflow+"|"+wall, RT=False)
        Vhat = TangentialFacetFESpace(mesh, order=order-1, dirichlet=inflow+"|"+wall+"|"+outflow) # , hide_highest_order_dc=True)
        Sigma = HCurlDiv(mesh, order = order-1, orderinner=order, discontinuous=True)

This file has been truncated. show original

Joachim

dong · June 25, 2020, 11:17pm

Thank you so much for your clarification. I think ngsolve will be more user-friendly if it has an L2-projection built-in function.