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Error bounds for full space-time splitting discretizations of semi-linear SPDEs -- with a focus on dG domain decompositions

Monika Eisenmann, Eskil Hansen, Marvin Jans

TL;DR

The paper develops a non-iterative domain-decomposition framework for parabolic SPDEs with multiplicative noise by embedding a Douglas–Rachford splitting into time integration and pairing it with a discontinuous Galerkin spatial discretization. It establishes a robust space-time convergence theory, deriving explicit error bounds under broad assumptions and detailing how these bounds adapt when A is self-adjoint and when the operator split commutes. The approach enables parallel computation and reduces the overhead of iterative subdomain solves, while remaining applicable to a broad class of semi-linear and quasi-linear SPDEs. Numerical experiments with a fully discretized domain-decomposition scheme validate the theoretical predictions and demonstrate the method’s practical effectiveness and even error distribution across the domain, as well as its potential for parallel scalability. Overall, the work provides a solid foundation for efficient, scalable SPDE solvers that leverage domain decomposition and DG discretization, with explicit convergence guarantees.

Abstract

We consider a fully discretized numerical scheme for parabolic stochastic partial differential equations with multiplicative noise. Our abstract framework can be applied to formulate a non-iterative domain decomposition approach. Such methods can help to parallelize the code and therefore lead to a more efficient implementation. The domain decomposition is integrated through the Douglas-Rachford splitting scheme, where one split operator acts on one part of the domain. For an efficient space discretization of the underlying equation, we chose the discontinuous Galerkin method as this suits the parallelization strategy well. For this fully discretized scheme, we provide a strong space-time convergence result. We conclude the manuscript with numerical experiments validating our theoretical findings.

Error bounds for full space-time splitting discretizations of semi-linear SPDEs -- with a focus on dG domain decompositions

TL;DR

The paper develops a non-iterative domain-decomposition framework for parabolic SPDEs with multiplicative noise by embedding a Douglas–Rachford splitting into time integration and pairing it with a discontinuous Galerkin spatial discretization. It establishes a robust space-time convergence theory, deriving explicit error bounds under broad assumptions and detailing how these bounds adapt when A is self-adjoint and when the operator split commutes. The approach enables parallel computation and reduces the overhead of iterative subdomain solves, while remaining applicable to a broad class of semi-linear and quasi-linear SPDEs. Numerical experiments with a fully discretized domain-decomposition scheme validate the theoretical predictions and demonstrate the method’s practical effectiveness and even error distribution across the domain, as well as its potential for parallel scalability. Overall, the work provides a solid foundation for efficient, scalable SPDE solvers that leverage domain decomposition and DG discretization, with explicit convergence guarantees.

Abstract

We consider a fully discretized numerical scheme for parabolic stochastic partial differential equations with multiplicative noise. Our abstract framework can be applied to formulate a non-iterative domain decomposition approach. Such methods can help to parallelize the code and therefore lead to a more efficient implementation. The domain decomposition is integrated through the Douglas-Rachford splitting scheme, where one split operator acts on one part of the domain. For an efficient space discretization of the underlying equation, we chose the discontinuous Galerkin method as this suits the parallelization strategy well. For this fully discretized scheme, we provide a strong space-time convergence result. We conclude the manuscript with numerical experiments validating our theoretical findings.

Paper Structure

This paper contains 19 sections, 31 theorems, 163 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem description
  3. Discretization
  4. Spatial discretization
  5. Temporal discretization
  6. Convergence analysis
  7. Basic estimates
  8. Convergence results
  9. Example: A fully discretized domain decomposition scheme
  10. Continuous problem
  11. Discontinuous Galerkin spatial discretization
  12. Full discretization
  13. Numerical experiments
  14. Semi-linear test example
  15. Quasi-linear test example
  16. ...and 4 more sections

Key Result

Theorem 2.6

Let Assumptions ass:A--ass:B be fulfilled. Then there exists a unique mild solution to Equation eq:SPDE up to modifications. Assuming that $\theta_{X_0} \in[\theta_{B},\theta_{B} + \frac{1}{2})$, the solution fulfills the following two regularity bounds and

Figures (2)

  • Figure 1: Left: Space convergence plot of the strong error at the final time for Experiment 1. Right: Time convergence plot of the strong error at the final time for Experiment 1.
  • Figure 2: Time convergence plot of the strong error at the final time for Experiment 2.

Theorems & Definitions (59)

  • Theorem 2.6
  • proof
  • Remark 3.2
  • Lemma 3.5
  • proof
  • Remark 3.6
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 49 more