Source-linked AI summary
Unconditionally optimal error estimates of a Crank--Nicolson Galerkin method for the nonlinear thermistor equations
Buyang Li, Weiwei Sun
TL;DR
Linearized semi-implicit analyses of nonlinear thermistor equations have often required time-step restrictions. This paper introduces an uncoupled, linearized Crank–Nicolson Galerkin method and obtains unconditionally optimal error estimates in L2 and H1 norms.
Problem
Error analyses of linearized semi-implicit schemes for nonlinear thermistor equations have required time-step restrictions, with prior optimal estimates reported under specified restrictions.
Source: §1 · p. 2Method
The method uses Crank–Nicolson for the temperature equation’s linear term, extrapolates conductivity, and splits errors via a corresponding time-discrete system.
Source: §1 · p. 3Results
The proposed method has unconditionally optimal error estimates in both L2 and H1 norms, and numerical results confirm the analysis.
Source: §1 · p. 3, §6 · p. 19Takeaways & Limitations
The authors state that the approach can extend to more general nonlinear parabolic PDEs and other time discretizations for unconditional convergence and optimal error estimates.
Source: §1 · p. 3Temporal error estimates hold for τ < τ0 beyond the initial steps, while at the initial steps they hold for any τ.
Source: §3 · p. 10
Abstract
from arXiv · showhide
This paper focuses on unconditionally optimal error analysis of an uncoupled and linearized Crank--Nicolson Galerkin finite element method for the time-dependent nonlinear thermistor equations in $d$-dimensional space, $d=2,3$. We split the error function into two parts, one from the spatial discretization and one from the temporal discretization, by introducing a corresponding time-discrete (elliptic) system. We present a rigorous analysis for the regularity of the solution of the time-discrete system and error estimates of the time discretization. With these estimates and the proved regularity, optimal error estimates of the fully discrete Crank--Nicolson Galerkin method are obtained unconditionally. Numerical results confirm our analysis and show the efficiency of the method.
1 Introduction
The introduction motivates removing time-step restrictions from error analysis of linearized schemes for nonlinear thermistor equations. It presents an uncoupled, linearized Crank–Nicolson Galerkin method based on splitting spatial and temporal errors through a time-discrete system.
1 Introduction
- The time-dependent thermistor model describes electric heating, with temperature u and temperature-dependent electric conductivity σ, in dimensions d = 2, 3.Source: §1 · p. 1
- Earlier thermistor analyses reported optimal L2 error estimates under time-step restrictions, including τ = O(h1/2) and τ = O(h3/2p) in cited settings.The introduction notes that such restrictions can require very small steps and make computations time-consuming.Source: §1 · p. 2
- Linearized semi-implicit schemes require boundedness estimates for the numerical solution, while induction with inverse inequalities imposes time-step conditions.Source: §1 · p. 2
- The authors present estimates without time-step conditions and say the approach may extend to more general nonlinear parabolic PDEs and other time discretizations.Source: §1 · pp. 2–3
- The proposed method applies Crank–Nicolson to the temperature equation’s linear term, extrapolates conductivity, and solves two uncoupled linear systems per step.Its error analysis separates spatial and temporal errors using a corresponding time-discrete elliptic system.Source: §1 · p. 3
2 The main result
The section presents an uncoupled, linearized Crank–Nicolson Galerkin method and splits its error into spatial and temporal parts using a time-discrete elliptic system. Under the stated solution assumptions, it establishes existence and uniqueness and gives an unconditional error estimate for the potential.
2 The main result
- The method uses Crank–Nicolson for the linear temperature term and extrapolation for nonlinear conductivity, requiring two uncoupled linear solves at each time step.The error is separated into spatial and temporal components through a corresponding time-discrete elliptic system.Source: §2 · pp. 4–5
- Under the stated solution assumptions, the fully discrete finite element system admits a unique solution.The theorem’s error estimates are stated for the fully discrete solution, with the potential evaluated at time levels tn−1/2 and tn.Source: §2 · pp. 4–5
- The time-discrete elliptic system is introduced to establish the regularity needed for the Galerkin finite element error estimates.The fully discrete system is viewed as its spatial discretization.Source: §2 · p. 5
- The potential error at integer time levels satisfies max over 1 ≤ n ≤ N of ||Φ_h^n − φ(·, t_n)||_{W^{1,12/5}} ≤ C(τ^2 + h).The estimate is stated after recovering the potential at time level tn.Source: §2 · pp. 5–6
3 Temporal error analysis
The temporal analysis establishes solvability and estimates for the time-discrete system, including regularity bounds for its temperature and potential. The stated bounds hold for sufficiently small time steps, while the initial-step bounds hold for any time step.
3 Temporal error analysis
- The analysis establishes existence and uniqueness for the time-discrete system and error bounds for the temporal error functions.The proof derives estimates for the temperature and potential errors using their error equations and truncation-error bounds.Source: §3 · pp. 6–7
- The time-discrete system has a unique solution with max_n ||U^n||_{H2}, max_n ||DτU^n||_{H2}, and ||Dτ bU^{1/2}||_{H2} bounded by C.The potential also satisfies max_n ||Φ^{n−1/2}||_{W^{2,12/5}} and max_n ||∇Φ^{n−1/2}||_{L^p} bounded by C for every 1 ≤ p < ∞.Source: §3 · p. 6
- The regularity estimates (3.1)–(3.3) are proved for τ < τ0, whereas they hold at the initial steps for any τ > 0.The small-step condition is part of the temporal regularity analysis.Source: §3 · p. 10
4 Spatial error analysis
This section estimates the spatial discretization error for the Galerkin finite element method applied to the time-discrete system. Its estimates contribute to Theorem 2.1 together with Theorem 3.1 and bounds (4.4) and (4.7).
4 Spatial error analysis
- The projections Π_h and R_h have O(h^2) Lp error bounds with h-weighted W1,p errors for the stated W2,p regularity ranges.Source: §4 · pp. 10–11
- Under the stated solution regularity assumptions, the fully discrete finite element system admits a unique solution.Source: §4 · p. 11
- Each time step requires solving two uncoupled linear discrete elliptic systems whose coefficient matrices are symmetric positive definite.Source: §4 · p. 11
- The spatial estimate bounds max_{0≤n≤N} ||e_h^n||_L2 + max_{0≤n≤N−1} ||η_h^(n+1/2)||_L12/5 by C.Source: §4 · p. 14
- Theorem 2.1 follows from Theorems 3.1 and 4.1 together with bounds (4.4) and (4.7).Source: §4 · p. 15
5 Numerical results
The numerical examples test spatial error rates for linear and quadratic finite elements, compare the results with the analysis, and examine stability at large time steps. The three-dimensional example also tests L2 convergence and large-step stability.
5 Numerical results
- The numerical tests use FEniCS and linear elements, with a uniform triangular partition in Example 4.1 and a uniform tetrahedral partition in Example 4.2.The latter has 6M3 tetrahedra and (M + 1)3 vertices.Source: §5 · pp. 15, 18
- For linear FEM in Example 4.1, L2 errors are proportional to h2 with τ = h.Table 1 reports the L2 errors for h = τ = 1/M.Source: §5 · p. 15, Table 1 · p. 16
- For quadratic FEM in Example 4.1, L2 errors are proportional to h3 with τ = h3/2.Table 2 reports the L2 errors for h = 1/M and τ = h3/2.Source: §5 · pp. 15–16, Table 2 · p. 16
- The scheme remains stable at large time steps in both examples, though the Example 4.1 results at τ = 20h seem not very accurate.Example 4.2 tests τ = h, 5h, and 10h; the reported results show unconditional stability.Source: §5 · pp. 16, 18
- The H1 error tests for linear and quadratic FEM use the stated time-step choices, and their results agree with the theoretical analysis.The tests use τ = h1/2 for linear FEM and τ = h3/2 for quadratic FEM.Source: §5 · p. 16, Table 3 · p. 17, Table 4 · p. 17
- In Example 4.2, L2 errors for both u and φ are proportional to h2 with τ = h.The computation uses linear elements on a uniform tetrahedral partition.Source: §5 · p. 18, Table 6 · p. 19
6 Conclusions
The paper presents an uncoupled, linearized Crank–Nicolson Galerkin method for time-dependent nonlinear thermistor equations in two and three dimensions. It proves unconditionally optimal error estimates in L2 and H1, and reports numerical results that confirm the analysis.
6 Conclusions
- The method yields unconditionally optimal error estimates in L2 and H1 for the d = 2, 3 nonlinear time-dependent thermistor equations, unlike existing analyses requiring time-step restrictions.The authors state that the numerical results confirm the analysis and show the method's efficiency; extension to other systems and approximations is proposed, while the analysis focuses on electric heating with linear finite elements.Source: §6 · p. 19