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Additive Schwarz methods for semilinear elliptic problems with convex energy functionals: Convergence rate independent of nonlinearity

Jongho Park

TL;DR

This paper shows that additive Schwarz methods for semilinear elliptic problems with convex energy functionals have convergence rates that are independent of the nonlinear term, effectively localizing nonlinearity within subdomain solves. By decomposing the energy into a smooth part $F_h$ and a nonsmooth part $G_h$ and employing a positivity-preserving nonlinear coarse interpolation $J_H$, the authors establish stable, scalable one- and two-level domain decomposition methods. The one-level analysis yields a rate bound with $C_0^2 \approx 1/(H\delta)$, while the two-level method achieves $C_0^2 \approx C_d(H,h)^2(1+H/\delta)$, making the two-level approach scalable with respect to $H/h$ and $H/\delta$. Numerical experiments on monomial, nonlinear Poisson–Boltzmann, and $L^1$-penalized problems corroborate the theory, showing uniform convergence rates across nonlinearities and clear scalability for the two-level method.

Abstract

We investigate additive Schwarz methods for semilinear elliptic problems with convex energy functionals, which have wide scientific applications. A key observation is that the convergence rates of both one- and two-level additive Schwarz methods have bounds independent of the nonlinear term in the problem. That is, the convergence rates do not deteriorate by the presence of nonlinearity, so that solving a semilinear problem requires no more iterations than a linear problem. Moreover, the two-level method is scalable in the sense that the convergence rate of the method depends on $H/h$ and $H/δ$ only, where $h$ and $H$ are the typical diameters of an element and a subdomain, respectively, and $δ$ measures the overlap among the subdomains. Numerical results are provided to support our theoretical findings.

Additive Schwarz methods for semilinear elliptic problems with convex energy functionals: Convergence rate independent of nonlinearity

TL;DR

This paper shows that additive Schwarz methods for semilinear elliptic problems with convex energy functionals have convergence rates that are independent of the nonlinear term, effectively localizing nonlinearity within subdomain solves. By decomposing the energy into a smooth part and a nonsmooth part and employing a positivity-preserving nonlinear coarse interpolation , the authors establish stable, scalable one- and two-level domain decomposition methods. The one-level analysis yields a rate bound with , while the two-level method achieves , making the two-level approach scalable with respect to and . Numerical experiments on monomial, nonlinear Poisson–Boltzmann, and -penalized problems corroborate the theory, showing uniform convergence rates across nonlinearities and clear scalability for the two-level method.

Abstract

We investigate additive Schwarz methods for semilinear elliptic problems with convex energy functionals, which have wide scientific applications. A key observation is that the convergence rates of both one- and two-level additive Schwarz methods have bounds independent of the nonlinear term in the problem. That is, the convergence rates do not deteriorate by the presence of nonlinearity, so that solving a semilinear problem requires no more iterations than a linear problem. Moreover, the two-level method is scalable in the sense that the convergence rate of the method depends on and only, where and are the typical diameters of an element and a subdomain, respectively, and measures the overlap among the subdomains. Numerical results are provided to support our theoretical findings.
Paper Structure (16 sections, 12 theorems, 89 equations, 3 figures, 4 tables, 1 algorithm)

This paper contains 16 sections, 12 theorems, 89 equations, 3 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model problem
  3. Semilinear elliptic problems
  4. Finite element discretization
  5. Additive Schwarz method
  6. Overlapping domain decomposition methods
  7. One-level method
  8. Two-level method
  9. Numerical results
  10. Monomial nonlinearity
  11. Nonlinear Poisson--Boltzmann equation
  12. L1-penalized problem
  13. Concluding remarks
  14. Proof of Proposition 2.1
  15. Proof of Theorem 2.4
  16. ...and 1 more sections

Key Result

Proposition 2.1

\newlabelProp:semilinear_convex0 In model and semilinear, suppose that the following hold: Then the map $u \mapsto \int_{\Omega} \phi (x, u(x)) \,dx$ is Gâteaux differentiable and semilinear_convex holds. Consequently, semilinear admits the convex variational formulation model.

Figures (3)

  • Figure 1: Decay of the relative energy error $\frac{E_h (u^{(n)}) - E_h (u_h)}{|E_h (u_h)|}$ in additive Schwarz methods for the monomial nonlinearity problem \ref{['monomial']} ($m = 3$, $\alpha = 10^1$).
  • Figure 2: Decay of the relative energy error $\frac{E_h (u^{(n)}) - E_h (u_h)}{|E_h (u_h)|}$ in additive Schwarz methods for the nonlinear Poisson--Boltzmann equation \ref{['PB']} ($\alpha = 10^0$).
  • Figure 3: Decay of the relative energy error $\frac{E_h (u^{(n)}) - E_h (u_h)}{|E_h (u_h)|}$ in additive Schwarz methods for the $L^1$-penalized problem \ref{['L1']} ($\alpha = 10$).

Theorems & Definitions (28)

  • Proposition 2.1
  • Proof 1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Theorem 2.5
  • Proof 2
  • Remark 2.6
  • Theorem 3.3
  • Theorem 4.1
  • ...and 18 more