Interior a posteriori error estimates for time discrete approximations of parabolic problems

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Numerical Solution to Parabolic PDE Using Implicit Finite Difference Approach

This paper examines an implicit Finite Difference approach for solving the parabolic partial differential equation (PDE) in one dimension. We consider the Crank Nicolson scheme which offers a better truncation error for both time and spatial dimensions as compared with the explicit Finite Difference method. In addition the scheme is consistent and unconditionally stable. One downside of implicit methods is the relatively high computational cost involved in the solution process, however this is compensated by the high level of accuracy of the approximate solution and efficiency of the numerical scheme. A physical problem modelled by the heat equation with Neumann boundary condition is solved using the Crank Nicolson scheme. Comparing the numerical solution with the analytical solution, we observe that the relative error increases sharply at the right boundary, however it diminishes as the spatial step size approaches zero. Keywords: Partial Differential Equation, Implicit Finite Difference, Crank Nicolson Schem

Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

Abstract. We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods. 1

Analysis for time discrete approximations of blow-up solutions of semilinear parabolic equations

We prove a posteriori error estimates for time discrete approximations, for semilinear parabolic equations with solutions that might blow up in finite time. In particular we consider the backward Euler and the Crank–Nicolson methods. The main tools that are used in the analysis are the reconstruction technique and energy methods combined with appropriate fixed point arguments. The final estimates we derive are conditional and lead to error control near the blow up time

An adaptive algorithm for the Crank–Nicolson scheme applied to a time-dependent convection–diffusion problem

AbstractAn a posteriori upper bound is derived for the nonstationary convection–diffusion problem using the Crank–Nicolson scheme and continuous, piecewise linear stabilized finite elements with large aspect ratio. Following Lozinski et al. (2009) [13], a quadratic time reconstruction is used.A space and time adaptive algorithm is developed to ensure the control of the relative error in the L2(H1) norm. Numerical experiments illustrating the efficiency of this approach are reported; it is shown that the error indicator is of optimal order with respect to both the mesh size and the time step, even in the convection dominated regime and in the presence of boundary layers