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In this article, Richardson extrapolation technique is employed to investigate the local ultraconvergence properties of Lagrange finite element method using piecewise polynomials of degrees k (k≥2) for the second order elliptic problem with inhomogeneous boundary. A sequence of special graded partition TNs are proposed and a new interpolation operator is introduced to achieve 2k order local ultraconvergence for the displacement and derivative. [ABSTRACT FROM AUTHOR]

In this article, we will investigate the local ultraconvergence of the displacement and derivative of finite element solution using piecewise polynomials of degrees k (k ≥ 2) by Richardson extrapolation operators P1k and P2k presented in He et al. for second‐order elliptic problems with singularity. Numerical results are provided to illustrate the theoretic findings. [ABSTRACT FROM AUTHOR]

A new fast numerical scheme is proposed for solving time-dependent coupled Burgers' equations. The idea of operator splitting is used to decompose the original problem into nonlinear pure convection subproblems and diffusion subproblems at each time step. Using Taylor's expansion, the nonlinearity in convection subproblems is explicitly treated by resolving a linear convection system with artificial inflow boundary conditions that can be independently solved. A multistep technique is proposed to rescue the possible instability caused by the explicit treatment of the convection system. Meanwhile, the diffusion subproblems are always self-adjoint and coercive at each time step, and they can be efficiently solved by some existing preconditioned iterative solvers like the preconditioned conjugate galerkin method, and so forth. With the help of finite element discretization, all the major stiffness matrices remain invariant during the time marching process, which makes the present approach extremely fast for the time-dependent nonlinear problems. Finally, several numerical examples are performed to verify the stability, convergence and performance of the new method. [ABSTRACT FROM AUTHOR]