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Global convergence of Newton's method for the regularized $p$-Stokes equations

Niko Schmidt

Abstract

The motion of glaciers can be simulated with the $p$-Stokes equations. Up to now, Newton's method to solve these equations has been analyzed in finite-dimensional settings only. We analyze the problem in infinite dimensions to gain a new viewpoint. We do that by proving global convergence of the infinite-dimensional Newton's method with Armijo step sizes to the solution of these equations. We only have to add an arbitrarily small diffusion term for this convergence result. We prove that the additional diffusion term only causes minor differences in the solution compared to the original $p$-Stokes equations under the assumption of some regularity. Finally, we test our algorithms on two experiments: A reformulation of the experiment ISMIP-HOM $B$ without sliding and a block with sliding. For the former, the approximation of exact step sizes for the Picard iteration and exact step sizes and Armijo step sizes for Newton's method are superior in the experiment compared to the Picard iteration. For the latter experiment, Newton's method with Armijo step sizes needs many iterations until it converges fast to the solution. Thus, Newton's method with approximately exact step sizes is better than Armijo step sizes in this experiment.

Global convergence of Newton's method for the regularized $p$-Stokes equations

Abstract

The motion of glaciers can be simulated with the -Stokes equations. Up to now, Newton's method to solve these equations has been analyzed in finite-dimensional settings only. We analyze the problem in infinite dimensions to gain a new viewpoint. We do that by proving global convergence of the infinite-dimensional Newton's method with Armijo step sizes to the solution of these equations. We only have to add an arbitrarily small diffusion term for this convergence result. We prove that the additional diffusion term only causes minor differences in the solution compared to the original -Stokes equations under the assumption of some regularity. Finally, we test our algorithms on two experiments: A reformulation of the experiment ISMIP-HOM without sliding and a block with sliding. For the former, the approximation of exact step sizes for the Picard iteration and exact step sizes and Armijo step sizes for Newton's method are superior in the experiment compared to the Picard iteration. For the latter experiment, Newton's method with Armijo step sizes needs many iterations until it converges fast to the solution. Thus, Newton's method with approximately exact step sizes is better than Armijo step sizes in this experiment.
Paper Structure (19 sections, 18 theorems, 83 equations, 6 figures, 1 table)

This paper contains 19 sections, 18 theorems, 83 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. The p-Stokes equations
  3. Variational formulation
  4. Regularization of the equation and the divergence-free space
  5. Equivalent minimization problem
  6. Existence and uniqueness of weak solutions in the divergence-free space
  7. Differentiability
  8. Infinite-dimensional Newton's Method
  9. Globalized Newton method
  10. Step size controls
  11. Global convergence
  12. Convergence to the non regularized problem
  13. Numerical experiment
  14. Numerical solvers and computational effort
  15. Computational domain
  16. ...and 4 more sections

Key Result

Lemma 2.2

Let $p\in (1,2)$. For all $P \in L^p(\Omega)^{N \times N}$ follows $S^p(P)\in L^{p'}(\Omega)^{N \times N}$ with the dual exponent $p'$.

Figures (6)

  • Figure 1: Considered domain in our experiment.
  • Figure 2: The velocity norms at the surface are shown in the left figure. All methods produce nearly the same surface velocity as the result.
  • Figure 3: Left: The relative norms of the Riesz isomorphism are visualized for different values of $\delta$. Right: The velocity norms at the surface are shown in the right figure.
  • Figure 4: The relative norms of the Riesz isomorphism are visualized for $\delta:=10^{-4}$.
  • Figure 5: The friction coefficient is $\tau:=10^7$. Left: The initial guess is the same as in experiment ISMIP-HOM $B$. Right: The initial guess has the additional term $\int_{\Gamma_b}\tau \boldsymbol{v_0}\cdot \boldsymbol{\phi}\, ds$.
  • ...and 1 more figures

Theorems & Definitions (49)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3: Function space
  • Definition 2.4: Divergence free space
  • Definition 2.5: Weak solution of the $p$-Stokes equations
  • Definition 2.6: Convex functional
  • Lemma 2.7
  • proof
  • Definition 2.8: Strict monotonicity, coercivity
  • ...and 39 more