[Trilinos-Users] [EXTERNAL] Transforming Laplacians of Intrepid FEM basis functions

Peterson, Kara J. kjpeter at sandia.gov
Mon Mar 21 12:54:14 EDT 2016

Hi Nate,

The transformation you included below does look correct, but there may be an easier way of implementing a stabilization term that involves a Laplacian than directly computing the second derivative of the basis functions.

Although I haven't actually implemented this myself, I understand that hyperviscosity-like stabilizations that involve Laplacians are usually applied using a weak notion of the Laplacian in finite elements. So given a test function  v

\int (\nabla \cdot \nabla u) v d\Omega  = \int \nabla u \cdot \nabla v d\Omega + boundary terms.

In this way the Laplacian of a variable can be written as M^{-1}K u for a mass matrix M and stiffness matrix K.

Also, if it would be helpful to you I do have examples of SUPG stabilization using Intrepid, but the implementations did not involve any explicit Laplacian terms.

Best,
Kara

________________________________
Kara Peterson
Computational Mathematics
Sandia National Laboratories
505.844.9372

From: Trilinos-Users <trilinos-users-bounces at trilinos.org<mailto:trilinos-users-bounces at trilinos.org>> on behalf of "Roberts, Nathan V." <nvroberts at alcf.anl.gov<mailto:nvroberts at alcf.anl.gov>>
Date: Monday, March 21, 2016 at 8:15 AM
To: "<trilinos-users at trilinos.org<mailto:trilinos-users at trilinos.org>>" <trilinos-users at trilinos.org<mailto:trilinos-users at trilinos.org>>
Subject: [EXTERNAL] [Trilinos-Users] Transforming Laplacians of Intrepid FEM basis functions

Hello all,

I'm implementing an SUPG formulation for convection-diffusion, and the stabilization term for this involves a Laplacian.  So, I need to take second-order derivatives of the FEM basis functions in Intrepid.  I see how to do this in reference space; I can apply OPERATOR_D2 and use the Intrepid::getDkEnumeration() to navigate the returned values.

My question is this: how do I appropriately transform the values to physical space?  I.e., the second-order analogue to the FunctionSpaceTools::HGRADtransformGRAD()?

Below, I offer what I've worked out thus far-I'm hoping that someone can (a) confirm that what I have is reasonable, and (b) comment on the simplest/best way to approach this in terms of what's available in Intrepid.

Mathematically, what HGRADtransformGRAD() is doing goes something like this.  If the physical function is u and the reference-space function is \hat{u}, and there is a vector transformation function x = x(\xi) from reference to physical space, then the gradient of u can be computed as:

This can be computed by multiplying the inverse of the Jacobian of the reference-to-physical map by the reference space gradient values; this is what HGRADtransformGRAD does.

Proceeding to the second derivatives needed for the Laplacian, we have

I believe the first term on the right hand side is computable in terms of the reference-space OPERATOR_D2 values and the inverse of the Jacobian applied twice (I would be glad to see example code if anyone has that to offer).  The second term requires second derivatives of the reference-to-physical map (i.e. the Hessian), as well as first derivatives of the reference-space function.

Is that correct?  Can anyone comment on the best way to implement this?

Thanks!
Nate

P.S. In case my pasted PDF graphics do not come through, the LaTeX for the first equation is:
\frac{\partial u}{\partial x_i} = \frac{\partial \hat{u}}{\partial \xi_k} \frac{\partial \xi_k}{\partial x_i}

and the second is:
\frac{\partial^2 u}{\partial x_i^2} = \frac{\partial^2 \hat{u}}{\partial \xi_k \partial \xi_l} \frac{\partial \xi_k}{\partial x_i} \frac{\partial \xi_l}{\partial x_i} + \frac{\partial \hat{u}}{\partial \xi_k} \frac{\partial^2 \xi_k}{\partial x_i^2}

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