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Register a new userConvergence of the Crank-Nicolson-Galerkin finite element method for a class of nonlocal parabolic systems with moving boundaries
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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.
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The aim of this paper is to establish convergence, properties and error bounds for the fully discrete solutions of a class of nonlinear systems of reaction-diffusion nonlocal type with moving boundaries, using the 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 a moving finite element method are investigated.
The aim of this paper is the numerical study of a class of nonlinear nonlocal degenerate parabolic equations. The convergence and error bounds of the solutions are proved for a linearized Crank-Nicolson-Galerkin finite element method with polynomial approximations of degree $kgeq 1$. Some explicit solutions are obtained and used to test the implementation of the method in Matlab environment.
The iterated Crank-Nicolson (ICN) method is a successful numerical algorithm in numerical relativity for solving partial differential equations. The $theta$-ICN method is the extension of the original ICN method where $theta$ is the weight when averaging the predicted and corrected values. It has better stability when $theta$ is chosen to be larger than 0.5, but the accuracy is reduced since the $theta$-ICN method is second order accurate only when $theta$ = 0.5. In this paper, we propose two modified $theta$-ICN algorithms that have second order of convergence rate when $theta$ is not 0.5, based on two different ways to choose the weight $theta$. The first approach employs two geometrically averaged $theta$s in two iterations within one time step, and the second one uses arithmetically averaged $theta$s for two consecutive time steps while $theta$ remains the same in each time step. The stability and second order accuracy of our methods are verified using stability and truncation error analysis and are demonstrated by numerical examples on linear and semi-linear hyperbolic partial differential equations and Burgers equation.
We propose a weak Galerkin(WG) finite element method for solving the one-dimensional Burgers equation. Based on a new weak variational form, both semi-discrete and fully-discrete WG finite element schemes are established and analyzed. We prove the existence of the discrete solution and derive the optimal order error estimates in the discrete $H^1$-norm and $L^2$-norm, respectively. Numerical experiments are presented to illustrate our theoretical analysis.
A new modified Galerkin / Finite Element Method is proposed for the numerical solution of the fully nonlinear shallow water wave equations. The new numerical method allows the use of low-order Lagrange finite element spaces, despite the fact that the system contains third order spatial partial derivatives for the depth averaged velocity of the fluid. After studying the efficacy and the conservation properties of the new numerical method, we proceed with the validation of the new numerical model and boundary conditions by comparing the numerical solutions with laboratory experiments and with available theoretical asymptotic results.
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