3
$\begingroup$

I am considering a two-grid finite element scheme for eigenvalue problems arising from PDEs. The toy problem I am addressing first is the 2D Laplace eigenvalue equation

$$\Delta u(\mathbf{x}) = \lambda u(\mathbf{x}), \quad \mathbf{x}\in\Omega$$ $$ u(\mathbf{x}) = 0, \quad \mathbf{x}\in\Gamma,$$

where $\Omega = [0,1]\times[0,1]$, with boundary denoted by $\Gamma$. I am using two uniform triangular meshes $\mathcal{T}_H$ and $\mathcal{T}_h$ (coarse and fine, respectively), such that $\mathcal{T}_h$ is obtained by one uniform refinement of $\mathcal{T}_H$. Denote the corresponding piecewise linear finite element spaces by $V_H$ and $V_h$, respectively. The associated matrix eigenvalue problems are then

$$A_H u_H = \lambda_H M_H u_H \quad \text{and} \quad A_h u_h = \lambda_h M_h u_h,$$

respectively. Here $A_H$ (resp. $A_h$) is the stiffness matrix given by:

$$ a_{ij} = a(\phi_i,\phi_j) = \int_{\Omega} \nabla\phi_i \nabla\phi_j dx dy,$$

and $M_H$ (resp. $M_h$) is the mass matrix given by:

$$ m_{ij} = (\phi_i,\phi_j) = \int_{\Omega} \phi_i \phi_j dx dy,$$

where $\{\phi_1,\dots,\phi_{N_H}\}$ (resp. $\{\phi_1,\dots,\phi_{N_h}\}$) are the standard linear basis functions.

For this particular problem $\{A_H,M_H\}$ and $\{A_h,M_h\}$ are symmetric positive-definite pencils, and consequently the eigenvectors of the coarse-grid problem are both $A_H$- and $M_H$-orthogonal (by which I mean, e.g., $u_{i,H}^{T} A_H u_{j,H} = 0$, whenever $i \neq j$), and similarly for the fine-grid problem.

Suppose I have two "exact" (machine precision) eigenpairs $(\lambda_{1,H},u_{1,H}) \in \mathbb{R}\times V_H$ and $(\lambda_{2,H},u_{2,H}) \in \mathbb{R}\times V_H$ for the coarse problem, so that $u_{1,H}$ and $u_{2,H}$ are $A_H$-orthogonal, and $\lambda_{i,H}$ taken to be the generalised Rayleigh quotient of $u_{i,H}$ with respect to $A_H$ and $B_H$. I want to construct two approximate eigenpairs, say $(\mu_{1,h},v_{1,h})\in \mathbb{R}\times V_h$ and $(\mu_{2,h},v_{2,h})\in \mathbb{R}\times V_h$, for the fine-grid problem by taking

$$ v_{i,h} = I(u_{i,H}) \quad i=1,2,$$

where $I(u_{i,H})$ denotes some interpolation of the coarse-grid solution onto the fine grid. $\lambda_{i,h}$ would then be taken to be the generalised Rayleigh quotient of $v_{i,h}$ with respect to $A_h$ and $B_h$.

Question: What assumptions are required on the interpolation method, in order to ensure that $v_{1,h}$ and $v_{2,h}$ are $A_h$-orthogonal?

My thoughts:

  1. Having looked at the simple 1D scenario, it seems intuitive to me that linear interpolation will yield the required orthogonality, due to the fact that the coarse-grid solution is piecewise linear (so, $I(u_{i,h})$ is essentially $u_{i,H}$ sitting in $V_h$). In a similar manner, I can deduce that the Rayleigh quotient approximations $\mu_{i,h}$ are exactly equal to $\lambda_{i,H}$.

  2. It does not seem obvious to me that the same will hold for higher degree interpolation. In particular, I am interested in using a cubic interpolant, as this gives far superior approximations, but my a priori error bounds are dependent on this orthogonality.

  3. Having implemented all of the above in MATLAB, both linear and cubic interpolation seem to preserve the orthogonality. This suggests that I am overlooking something with the cubic interpolation.

$\endgroup$
  • $\begingroup$ What is M? Is it a projection/prolongation operator? (don't recall which meant what)... $\endgroup$ – Emil Apr 24 '17 at 21:24
  • $\begingroup$ @Emil M denotes the mass matrix in this case. Sorry, this was not clear in my post - I have edited it to include the proper definitions of both A and M. $\endgroup$ – Kino Apr 24 '17 at 21:55
0
$\begingroup$

Actually, if you consider $y_{h,i} = P y_{H,i}$ and take the scalar product $$ (y_{h,i}, y_{h,j})_{A_h} = y_{H,i}^T P^T A_h P y_{H,j} = 0 $$ then you see that fine-grid vectors will be always orthogonal if you take coarse-grid matrix $A_H$ to be related to fine-grid matrix by $$ A_H = P^T A_h P $$ which is quite a standard thing to be assumed in multigrid methods. So, it is more about "how to define properly coarse-grid operator after choosing the interpolation" than "how to define interpolation for given $A_h$ and $A_H$".

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.