[Paper Review] Fully Adaptive Newton-Galerkin Time Stepping Methods for Singularly Perturbed Parabolic Evolution Equations

Mario Amrein, Thomas P. Wihler|arXiv (Cornell University)|Oct 2, 2015

Differential Equations and Numerical Methods13 references3 citations

TL;DR

This paper presents a fully adaptive Newton-Galerkin time stepping method for singularly perturbed semilinear parabolic equations with robust error control. By combining Newton linearization, adaptive backward Euler time stepping, and spatial finite element refinement, the method achieves $\varepsilon$-robust convergence even in the presence of boundary layers and spikes, validated through numerical experiments with varying perturbation parameters.

ABSTRACT

In this paper we develop an adaptive procedure for the numerical solution of semilinear parabolic problems, with possible singular perturbations. Our approach combines a linearization technique using Newton's method with an adaptive discretization-which is based on a spatial finite element method and the backward Euler time stepping scheme-of the resulting sequence of linear problems. Upon deriving a robust a posteriori error analysis, we design a fully adaptive Newton-Galerkin time stepping algorithm. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.

Motivation & Objective

• To develop a reliable and efficient numerical method for solving semilinear parabolic problems with small diffusion coefficients ($\varepsilon \ll 1$), which exhibit boundary layers, spikes, or shocks.
• To address the challenge of robust numerical approximation in the singularly perturbed regime, where standard methods fail due to loss of stability or accuracy.
• To design an adaptive algorithm that dynamically balances linearization, time, and spatial discretization errors for optimal efficiency.
• To ensure the error estimator remains robust as $\varepsilon \to 0$, enabling reliable error monitoring across all perturbation regimes.
• To validate the method’s performance on benchmark problems with known singular behavior, including nonlinear source terms and blow-up solutions.

Proposed method

• Newton’s method is applied to linearize the semilinear parabolic problem, transforming it into a sequence of linear evolution problems.
• The linearized problems are discretized in time using the backward Euler scheme and in space using $\mathbb{P}_1$-finite elements.
• An a posteriori error analysis is derived that decomposes the total residual into three computable components: linearization, time, and space discretization errors.
• The error estimator is constructed to be $\varepsilon$-robust by leveraging techniques from singularly perturbed elliptic problems and linear parabolic theory.
• A fully adaptive algorithm is designed to refine time steps or spatial meshes based on the dominant error component at each iteration.
• The adaptive strategy dynamically adjusts the Newton step size, time step, and spatial mesh based on residual indicators to ensure efficiency and robustness.

Experimental results

Research questions

• RQ1 Can a fully adaptive Newton-Galerkin method be constructed that maintains robustness with respect to the singular perturbation parameter $\varepsilon$ in semilinear parabolic problems?
• RQ2 How can the total error be decomposed into distinct contributions from linearization, time discretization, and spatial discretization for effective adaptive control?
• RQ3 Does the proposed adaptive algorithm reliably resolve boundary layers and interior spikes in the singularly perturbed regime as $\varepsilon \to 0$?
• RQ4 Can the error estimator provide reliable efficiency indices that remain bounded and stable across different values of $\varepsilon$?
• RQ5 What is the convergence behavior of the estimated error over time, particularly in problems with blow-up or rapid dynamics?

Key findings

• The proposed adaptive algorithm achieves $\varepsilon$-robust convergence, with efficiency indices remaining bounded across all tested values of $\varepsilon = 10^{-p}$ for $p = 1,\dots,5$.
• The estimated error grows at a rate of $\mathcal{O}(t_n^{1/2})$ over time, consistent with theoretical expectations from the error estimator.
• The method successfully resolves sharp boundary layers and interior spikes in problems with $\varepsilon = 10^{-5}$ and $\varepsilon = 10^{-3}$, respectively.
• In the blow-up problem with $\beta = 4$, the adaptive procedure accurately captures the developing spike near $x=2$ as time evolves.
• The log/log plot of the estimated error shows a slope of $1/2$, confirming the theoretical error growth rate and validating the reliability of the error estimator.
• The adaptive strategy effectively balances linearization, time, and spatial refinement, leading to optimal computational complexity and robust performance.