[Paper Review] Convergence of the Crank-Nicolson-Galerkin finite element method for a class of nonlocal parabolic systems with moving boundaries

Rui M.P. Almeida, José C. M. Duque+2 more|arXiv (Cornell University)|Jan 31, 2014

Differential Equations and Numerical Methods21 references3 citations

TL;DR

This paper establishes optimal convergence rates for a linearized Crank-Nicolson-Galerkin finite element method applied to nonlinear, nonlocal parabolic systems with moving boundaries. By using a coordinate transformation to fix the domain and employing polynomial basis functions of arbitrary degree, the method achieves optimal error bounds in both space and time, validated through numerical experiments showing high accuracy and agreement with adaptive moving finite element methods.

ABSTRACT

The aim of this paper is to establish the convergence and error bounds to the fully discrete solution for a class of nonlinear systems of reaction-diffusion nonlocal type with moving boundaries, using a linearized Crank-Nicolson-Galerkin finite element method with polynomial approximations of any degree. A coordinate transformation which fixes the boundaries is used. Some numerical tests to compare our Matlab code with some existing moving finite elements methods are investigated.

Motivation & Objective

• To develop a robust numerical method for solving nonlinear, nonlocal parabolic systems with moving boundaries, which arise in population dynamics and heat transfer.
• To establish rigorous convergence and error bounds for a fully discrete linearized Crank-Nicolson-Galerkin finite element method with arbitrary-order polynomial approximations.
• To handle nonlocal diffusion terms dependent on global integrals of the solution, ensuring stability and accuracy in time and space.
• To compare the proposed method with existing moving finite element methods in terms of accuracy and convergence behavior.
• To validate theoretical results through numerical simulations with varying mesh sizes, time steps, and parameter tolerances.

Proposed method

• A coordinate transformation is applied to map the non-cylindrical, moving-boundary domain into a fixed computational domain, enabling standard finite element discretization.
• The Crank-Nicolson scheme is used for temporal discretization, providing second-order accuracy and unconditional stability in time.
• Galerkin finite element method with piecewise polynomial basis functions of arbitrary degree is employed for spatial discretization.
• A linearization strategy is applied to the nonlinear system at each time step, solving a sequence of linear systems to improve computational efficiency.
• The method is implemented in MATLAB, with error norms computed in the L2(α(T), β(T)) space at final time T.
• Numerical comparisons are conducted with a moving finite element method (MFEM) from prior work, using identical problem setups and solution metrics.

Experimental results

Research questions

• RQ1 What is the optimal convergence rate of the linearized Crank-Nicolson-Galerkin finite element method for nonlocal parabolic systems with moving boundaries?
• RQ2 How does the method perform in comparison to adaptive moving finite element methods in terms of accuracy and error control?
• RQ3 Can the method achieve optimal error bounds in both space (h) and time (k) for arbitrary polynomial degrees?
• RQ4 How do varying diffusion coefficients and nonlocal terms affect the stability and convergence of the numerical solution?
• RQ5 What is the impact of the coordinate transformation on the conditioning and accuracy of the discrete system?

Key findings

• The method achieves optimal convergence rates in both space and time, with theoretical error bounds confirmed numerically for polynomial degrees 2 and 3.
• Numerical experiments show convergence order approximately 2 in space (h) and 2 in time (k), consistent with the theoretical prediction in Theorem 4.1.
• For a fixed mesh and time step, the error decreases significantly with smaller δ (tolerance in the Newton-Raphson solver), confirming robustness of the linearization process.
• In direct comparison with the moving finite element method (MFEM), the proposed method achieves comparable or superior accuracy, with errors reduced by up to two orders of magnitude at later times.
• The numerical results for both test cases show excellent agreement with the theoretical convergence behavior and with results from [20], validating the method’s reliability.
• The method maintains high accuracy even in regions of high solution gradient, with the error remaining small despite the moving boundary and nonlocal nonlinearity.