Convergent adaptive iterative schemes for solving multi-physics problems

TL;DR

This work addresses the challenge of solving nonlinear, strongly coupled multi-physics problems by developing a practical adaptive iterative framework based on cheap a posteriori estimators. It derives iteration-dependent norms and computable estimators to predict the success or failure of linearization or splitting schemes, enabling adaptive switching between methods and adaptive tuning of stabilization parameters and time steps. The approach is demonstrated on three porous-media problems—two-phase flow, surfactant transport, and quasi-static Biot poroelasticity—showing that switching and adaptive stabilization or time stepping markedly improves robustness and efficiency over fixed schemes. The framework provides a general, computationally light toolkit for robustly solving complex multi-physics systems with adaptive control informed by incremental-error estimators, offering practical impact for large-scale simulations and engineering applications.

Abstract

In this paper, we derive a practical, general framework for creating adaptive iterative (linearization or splitting) algorithms to solve multi-physics problems. This means that, given an iterative method, we derive \textit{a posteriori} estimators to predict the success or failure of the method. Based on these estimators, we propose adaptive algorithms, including adaptively switching between methods, adaptive time-stepping methods, and the adaptive tuning of stabilization parameters. We apply this framework to two-phase flow in porous media, surfactant transport in porous media, and quasi-static poroelasticity.

Figures (7)

• Figure 1: Illustration of different adaptive iterative algorithms based on a posteriori error estimators. Here $\eta_{\!_{\,i\to j}}$ represents the bound on the incremental error $\eta_{\!_{\,inc,i}}$ using the iterate of scheme $i$ to predict the error at the next iteration computed using scheme $j$.
• Figure 2: Test case: two-phase flow - The ratio between $\eta_{\!_{1\to 1}}/\eta_{\!_{inc,1}}$ at each iteration, i.e. the criteria for decreasing or increasing $L$ in the adaptive algorithm for the first time step when $\gamma=0.7$. Green indicates a decrease and red an increase of the stabilization parameter. The dashed line is $C_{\rm tol}=1$.
• Figure 3: Test case: two-phase flow - Evolution of switching indicators for the $L_2-N$ scheme for $h=\sqrt{2}/40$ and $\tau=0.1$ when $\gamma=0.5$. The effectivity indices \ref{['eq: eff ind']} corresponding to the Newton iterations ($\eta_{\!_{2\to 2}}$) are also plotted. The dashed line is $C_{\rm tol}=1$.
• Figure 4: Test case: surfactant transport - Total number of iterations at $T=1$ for $\tau=0.1$ and varying mesh size. For $N/\tau$ this means that the initial time step size is $\tau=0.1$. The number in blue parentheses corresponds to (number of L-scheme iterations/number of Newton iterations). The pink and orange parentheses correspond to (successful Newton iterations/unsuccessful Newton iterations). The L-scheme converges for all mesh sizes but uses more than 1000 iterations, and Newton's method converges when $\tau=0.0025$ on the coarsest mesh, with a total number of iterations larger than 1000 iterations; therefore, they are omitted from the plot. Newton diverges on finer meshes unless an even smaller time step is chosen.
• Figure 5: Test case: surfactant transport - Number of iterations at each time step for $h=2/\sqrt{60}$ with initial time step size $\tau=0.1$ for the adaptive time-stepping algorithm. The parentheses correspond to (successful Newton iterations/unsuccessful Newton iterations).

Theorems & Definitions (25)

• Definition 3.1: Residual two-phase flow
• Definition 3.2: Iteration dependent norm for L-scheme
• Definition 3.3: Iteration dependent norm for Newton's method
• Lemma 3.1: L-scheme to Newton estimator
• proof
• Lemma 3.2: Newton to Newton estimator
• proof
• Lemma 3.3: L-scheme to L-scheme estimator
• proof
• Remark 1: Extension to mixed formulations