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Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations
Buyang Li, Weiwei Sun
TL;DR
The paper addresses time-step restrictions in error analysis for linearized semi-implicit Galerkin methods applied to nonlinear parabolic equations. It introduces an error splitting through a corresponding time-discrete system and proves optimal L2- and H1-norm estimates without a time-step restriction for nonlinear Joule heating equations. The approach is also described as applicable to broader nonlinear parabolic systems and related discretizations.
Problem
Error analyses for linearized semi-implicit Galerkin schemes often require time-step restrictions to bound numerical solutions or errors in strong norms.
Method
The paper splits the error into temporal and spatial parts using a corresponding time-discrete parabolic system before analyzing the Galerkin approximation.
Results
The linearized semi-implicit Euler Galerkin method achieves optimal error estimates in both L2 and H1 norms without any time-step restriction for the nonlinear Joule heating system.
Takeaways & Limitations
The analysis can be applied to more general nonlinear parabolic systems and other linearized time discretizations for which prior analyses often impose step-size restrictions.
Takeaways & Limitations
The paper considers only linear Galerkin finite elements and focuses on the electric heating model, although extensions are indicated.
Abstract
from arXiv · showhide
This paper is concerned with the time-step condition of commonly-used linearized semi-implicit schemes for nonlinear parabolic PDEs with Galerkin finite element approximations. In particular, we study the time-dependent nonlinear Joule heating equations. We present optimal error estimates of the semi-implicit Euler scheme in both the $L^2$ norm and the $H^1$ norm without any time-step restriction. Theoretical analysis is based on a new splitting of the error and precise analysis of a corresponding time-discrete system. The method used in this paper can be applied to more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations for which previous works often require certain restriction on the time-step size $τ$.
1 Introduction
Linearized semi-implicit schemes reduce each time step to a linear solve but their error analyses often impose time-step restrictions to control strong norms. This paper develops an error splitting and time-discrete analysis that yields optimal Galerkin estimates without such restrictions for the nonlinear Joule heating system.
- Time-step challenge: Linearized semi-implicit schemes require only linear systems per time step, but error analyses often need strong-norm bounds that lead to time-step restrictions.Fully implicit schemes avoid such restrictions but require nonlinear solves, while explicit schemes face severely restricted step sizes.
- Scope: The approach is presented as applicable to more general nonlinear parabolic systems and other linearized time discretizations.The paper’s analysis is motivated by restrictions appearing across several classes of nonlinear parabolic equations.
- Related work: Previous Joule heating analyses obtained optimal L2 estimates under restrictions such as τ = O(hd/6) in three dimensions.
- Approach: The paper splits numerical error into temporal and spatial components through a corresponding time-discrete parabolic system.The resulting finite-element error bounds depend on h but not τ, provided suitable regularity of the time-discrete solution is established.
- Main contribution: For the nonlinear Joule heating model, the linearized semi-implicit backward Euler scheme with standard Galerkin approximation achieves optimal error estimates in both L2 and H1 norms without time-step restriction.The analysis targets the temperature-dependent conductivity system describing electric heating of a conducting body.
2 Galerkin methods and main results
The paper formulates a linearized semi-implicit Euler scheme with standard Galerkin finite elements for the nonlinear Joule heating system and analyzes its time-discrete and fully discrete errors.
- The spatial discretization uses linear Galerkin finite element spaces constructed over a regular triangulation or tetrahedral mesh of size h.
- The fully discrete scheme seeks finite element approximations U_h^n and Φ_h^n satisfying the discrete weak formulation at each time level.
- The analysis assumes solution regularity involving spatial W^{2,12/5} bounds, time derivatives in L^2(H^1), and Hölder control of gradients.
- Under the stated regularity and sufficiently small τ and h, the finite element system admits a unique solution, with constants independent of n, h, and τ.
- The corresponding time-discrete system defines U^n and Φ^n through a semi-implicit backward Euler treatment of the Joule heating equations.
- The proof uses a new error splitting together with projection operators and estimates for the time-discrete parabolic system.
3 Error estimates
The analysis splits the error through a corresponding time-discrete system, establishing regularity and then deriving fully discrete Galerkin error estimates for the nonlinear Joule heating equations. The resulting approach removes the prior time-step restriction from the optimal error bound.
- 3.1 The time-discrete solution: The proof first establishes existence, uniqueness, regularity, and error bounds for the time-discrete system before analyzing the Galerkin finite element error.The time-discrete analysis uses estimates, maximum-principle arguments, elliptic regularity, induction, and Gronwall’s inequality.
- 3.2 The fully-discrete finite element solution: The fully discrete error is split between the exact solution, the time-discrete solution, and the Galerkin finite element solution.This splitting separates temporal discretization error from the spatial finite element error.
- 3.2 The fully-discrete finite element solution: Theorem 3.2 establishes unique solvability of the fully discrete system for sufficiently small h and τ, and Theorem 2.1 follows from Theorems 3.1 and 3.2.The fully discrete result combines the time-discrete estimates with the finite element error analysis.
- 3.2 The fully-discrete finite element solution: Previous analyses required τ ≤ k0h^(d/6) when an induction argument used a preliminary bound for the exact-to-fully-discrete error.The restriction arose from estimating the fully discrete error directly through an inverse inequality.
- 3.2 The fully-discrete finite element solution: The new induction assumption instead controls the difference between the time-discrete and fully discrete solutions.This permits the optimal fully discrete error bound to be proved unconditionally with respect to the time-step size.
- 3.2 The fully-discrete finite element solution: At each time step, the Galerkin scheme requires solving two uncoupled linear systems whose coefficient matrices are symmetric and positive definite.Existence and uniqueness of the Galerkin finite element solution therefore follow directly under the stated assumption.
4 Conclusions
The paper establishes optimal error estimates and unconditional stability for linearized semi-implicit Galerkin schemes applied to three-dimensional nonlinear Joule heating equations. Its error-splitting analysis avoids time-step restrictions, while the conclusions note scope boundaries and extensions.
- The approach obtains optimal error estimates and unconditional stability for linearized semi-implicit Galerkin schemes in three-dimensional nonlinear Joule heating equations.
- A new error splitting separates time and spatial directions, enabling strong-norm control through induction and inverse inequalities without restricting the time-step size.
- The analysis is presented as extendable to other nonlinear parabolic systems, although this paper focuses on the electric heating model.
- The study considers only linear Galerkin finite element approximations, while high-order extensions are described as similar.The treatment also assumes g is defined in the domain; boundary-defined g requires incorporating boundary terms, with optimal estimates still expected without time-step restrictions.