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Two-level Nonlinear Preconditioning Methods for Flood Models Posed on Perforated Domains

Miranda Boutilier, Konstantin Brenner, Victorita Dolean

Abstract

This article focuses on the numerical solution of the Diffusive Wave equation posed in a domain containing a large number of polygonal perforations. These numerous perforations represent structures in urban areas, and this problem is used to model urban floods. This article relies on the work done in a previous article by the same authors, in which we introduced low-dimensional coarse approximation space for the linear Poisson equation based on a coarse polygonal partitioning of the domain. Similarly to other multi-scale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. We show that this coarse space also lends itself to the linearized Diffusive Wave problem that is obtained at each iteration of Newton's method. Furthermore, we present nonlinear preconditioning techniques, including a Two-level RASPEN and Two-step method, which can significantly reduce the number of iterations compared to the traditional Newton's method. For all proposed methods, we provide a discussion regarding the complexity of the algorithms. Numerical examples illustrating the performance of the proposed methods include a large-scale test case based on topographical data from the city of Nice.

Two-level Nonlinear Preconditioning Methods for Flood Models Posed on Perforated Domains

Abstract

This article focuses on the numerical solution of the Diffusive Wave equation posed in a domain containing a large number of polygonal perforations. These numerous perforations represent structures in urban areas, and this problem is used to model urban floods. This article relies on the work done in a previous article by the same authors, in which we introduced low-dimensional coarse approximation space for the linear Poisson equation based on a coarse polygonal partitioning of the domain. Similarly to other multi-scale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. We show that this coarse space also lends itself to the linearized Diffusive Wave problem that is obtained at each iteration of Newton's method. Furthermore, we present nonlinear preconditioning techniques, including a Two-level RASPEN and Two-step method, which can significantly reduce the number of iterations compared to the traditional Newton's method. For all proposed methods, we provide a discussion regarding the complexity of the algorithms. Numerical examples illustrating the performance of the proposed methods include a large-scale test case based on topographical data from the city of Nice.
Paper Structure (17 sections, 53 equations, 11 figures, 11 tables, 5 algorithms)

This paper contains 17 sections, 53 equations, 11 figures, 11 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Discretization of Diffusive Wave equation
  3. Multiscale Coarse space
  4. Linear Multi-domain Solution Methods
  5. Nonlinear Multi-domain Solution Methods
  6. Preconditioning the linearized Newton system
  7. Nonlinear Restrictive Additive Schwarz Iteration
  8. One-level RASPEN
  9. Two-level RASPEN
  10. Two-step method
  11. Anderson Acceleration of Coarse Two-step Method
  12. Complexities/Costs of Fine-scale Methods
  13. Numerical Results
  14. Porous Medium Equation on L-shaped Domain
  15. Porous Medium Equation on Large Urban Domain
  16. ...and 2 more sections

Figures (11)

  • Figure 1: Coarse cell $\Omega_j$, nonoverlapping skeleton $\Gamma$ (blue lines), and coarse grid nodes $\mathbf{x}_s \in {\cal V}$ (red dots). Coarse grid nodes are located at $\overline{\Gamma} \cap \partial \Omega_S.$
  • Figure 2: Finite element solution of \ref{['eq:pme']} for Example 1 on the chosen L-shaped domain.
  • Figure 3: Example 1, convergence curves. Top: $N=3$ (left), $N=5$ (right). Bottom: $N=9$.
  • Figure 4: Finite element solution of \ref{['eq:pmeex3']} for Example 2 on the chosen urban model domain.
  • Figure 5: Example 2, convergence curves. Solid lines correspond to an initial guess of $\mathbf{u}^0=1$, dotted lines correspond to an initial guess of $\mathbf{u}^0=0$. Top: $N=2 \times 2$ (left), $N=4 \times 4$ (right). Bottom: $N=8 \times 8$ (left), $N=16 \times 16$ (right).
  • ...and 6 more figures

Theorems & Definitions (3)

  • Definition 5.1: An NRAS iteration
  • Remark 5.1: NRAS is RAS when applied to a linear problem
  • Remark 5.2: Exact computation of the Jacobian