A Domain Decomposition Method for Nonconforming Finite Element Approximations of Eigenvalue Problems

Abstract

Since the nonconforming finite elements (NFEs) play a significant role in approximating PDE eigenvalues from below, this paper develops a new and parallel two-level preconditioned Jacobi-Davidson (PJD) method for solving the large scale discrete eigenvalue problems resulting from NFE discretization of 2mth $(m=1,2)$ order elliptic eigenvalue problems. Combining a spectral projection on the coarse space and an overlapping domain decomposition (DD), a parallel preconditioned system can be solved in each iteration. A rigorous analysis reveals that the convergence rate of our two-level PJD method is optimal and scalable. Numerical results supporting our theory are given.

Appendix A

The algorithm based on NFE for computing multiple and clustered eigenvalues is put in the following. For our earlier work based on CFEs, see [12] for details.

Algorithm A Two-Level PJD Algorithm for Multiple Eigenvalues
${\mathbf {Step\ 1}}$ Solve a coarse eigenvalue problem:
$A^{H}u_{i}^{H}=\lambda _{i}^{H}u_{i}^{H},\ \ \ b(u_{i}^{H},u_{j}^{H})=\delta _{ij},\ \ \ \ i,j=1,2,\cdots ,s.$
Set ${\bar{u}}_{i}^{0}=\frac{I_{H}^{h}u_{i}^{H}}{||I_{H}^{h}u_{i}^{H}||_{b}}$, ${\bar{U}}^{0}=\textrm{span}\{{\bar{u}}_{i}^{0}\}_{i=1}^{s}.$
Solve a small dimensional eigenvalue problem:
$a(u_{i}^{0},v)=\lambda _{i}^{0}b(u_{i}^{0},v)\ \ \ \ {\text{for all}} \ v\in {\bar{U}}^{0},\ b(u_{i}^{0},u_{j}^{0})=\delta _{ij},\ i,j=1,2,\cdots ,s.$
Set $U^{0}=\textrm{span}\{u_{i}^{0}\}_{i=1}^{s}, \Lambda ^{0}=\{\lambda _{i}^{0}\}_{i=1}^{s}$, $W^{0}=U^{0}$.
${\mathbf{Step\ 2}}$ For $k=0, 1, 2, \cdots ,$ solve parallel preconditioned systems:
$t_{i}^{k+1}=(I-Q_{U^{k}})(B_{i}^{k})^{-1}r_{i}^{k}=(B_{i}^{k})^{-1}r_{i}^{k}-\sum _{i=1}^{s}b((B_{i}^{k})^{-1}r_{i}^{k},u_{i}^{k})u_{i}^{k},$
where $r_{i}^{k}=\lambda _{i}^{k}u_{i}^{k}-A^{h}u_{i}^{k}$ and $(B_{i}^{k})^{-1}$ is defined in (A1).
${\mathbf{Step\ 3}}$ Solve the first s eigenpairs in $W^{k+1}$:
$a(u_{i}^{k+1},v)=\lambda _{i}^{k+1}b(u_{i}^{k+1},v)\ \ \ \ {\text{for all}} \ v\in W^{k+1},\ b(u_{i}^{k+1},u_{j}^{k+1})=\delta _{ij},$
where $\ i,j=1,2,\cdots ,s$, $W^{k+1}=W^{k}+\text {span}\{t_{i}^{k+1}\}_{i=1}^{s}$. Set $U^{k+1}=\text {span}\{u_{i}^{k+1}\}_{i=1}^{s},\ \Lambda ^{k+1}=\{\lambda _{i}^{k+1}\}_{i=1}^{s}.$
${\mathbf {Step\ 4}}$ If $\sum _{i=1}^{s}|\lambda _{i}^{k+1}-\lambda _{i}^{k}|<tol$, return $(\Lambda ^{k+1},U^{k+1})$. Otherwise, go to ${\mathbf{Step\ 2}}$.

The preconditioner for $A^{h}-\lambda _{i}^{k}$ is written as

$$\begin{aligned} (B_{i}^{k})^{-1}=I_{H}^{h}(A^{H}-\lambda _{i}^{k})^{-1}Q_{s+1}^{H}(I_{H}^{h})^{*} +\sum _{l=1}^{N}(A^{(l)}-\lambda _{i}^{k})^{-1}Q^{(l)}, \end{aligned}$$

(A1)

where $V_{CR}^{H}= U_{s}^{H}\oplus U_{s+1}^{H}$ with $U_{s}^{H}$ denoting the span of the first s discrete eigenvectors of $A^{H}$, and $Q_{s+1}^{H}$ is an orthogonal projector from $V_{CR}^{H}$ onto $U_{s+1}^{H}$ with respect to $b(\cdot ,\cdot )$.