Augmented Lagrangian preconditioners for fictitious domain formulations of elliptic interface problems

Abstract

We present a novel augmented Lagrangian (AL) preconditioner for the solution of linear systems arising from finite element discretizations of elliptic interface problems with jump coefficients. The method is based on the Fictitious Domain with Distributed Lagrange Multipliers formulation and it is designed to improve the convergence of the Flexible Generalized Minimal Residual (FGMRES) method in the presence of large coefficient jumps. To reduce the computational cost, we also introduce a cheaper block-triangular variant of the preconditioner. We prove eigenvalue clustering for the ideal AL preconditioner and study the limiting behavior of the spectrum for the modified variant in terms of parameters and the size of the jumps. Numerical experiments on different immersed geometries confirm mesh-independent iteration counts and robustness over large coefficient jumps, with substantial reductions in wall-clock time for the modified approach.

Figures (11)

• Figure 1: Top row: the initial configuration for Problem \ref{['prob:elliptic_interface']}, involving the domains $\Omega_1$ and $\Omega_2$. Bottom row: the fictitious domain reformulation, where the immersed domain $\Omega_2$ is superimposed on $\Omega$. The solution $u$ is defined in the whole $\Omega$, while $u_2$ is defined only in $\Omega_2$. The fictitious contribution is then subtracted at the variational level on the domain $\Omega_2$.
• Figure 2: $\beta=1$, $\beta_2 = 100$. Spectrum of the original system matrix $\mathcal{A}_{\gamma}$ (top row, in magenta) and $\mathcal{P}_{\gamma}^{-1}\mathcal{A}_{\gamma}$ (bottom row, in green) for increasing values of the augmentation parameter $\gamma$.
• Figure 3: $\beta=1$, $\beta_2 = 10^6$. Spectrum of the original system matrix $\mathcal{A}_{\gamma}$ (top row, in magenta) and $\mathcal{P}_{\gamma}^{-1}\mathcal{A}_{\gamma}$ (bottom row, in green) for increasing values of the augmentation parameter $\gamma$. Notice the different scale in the x-axis of the first row.
• Figure 4: $\beta=1$, $\beta_2 = 100$. Spectrum of the lower block \ref{['eqn:lower_block']} for increasing $\gamma_1$ and decreasing $\gamma_2$.
• Figure 5: Numerical solutions when $\beta_2=10^7$ for the square and ball geometries.

Theorems & Definitions (26)

• Proposition 1: AURICCHIO201536, Prop. 1
• Proposition 2: Discrete ellipticity on the kernel
• Proposition 3: Discrete inf-sup condition
• Remark 1: Singularity of $\mathsf{A_2}$
• Remark 2: Other mixed discretizations
• Remark 3: Case $V_{2,h} \ne \Lambda_h$
• Lemma 1
• Theorem 1: Spectrum of preconditioned matrix
• proof
• Lemma 2: Spectral equivalence with $h$-scaled mass matrix