论文标题
非线性抛物线方程的线性变量时端BDF2方案的无条件最佳误差估计值
Unconditionally optimal error estimate of a linearized variable-time-step BDF2 scheme for nonlinear parabolic equations
论文作者
论文摘要
在本文中，我们考虑了一个线性化的可变时间阶段两步向后分化公式（BDF2）方案，用于求解非线性抛物线方程。该方案是通过使用可变的时间键BDF2进行线性项来构建的，并在空间中与Galerkin有限元法（FEM）结合的非线性项的牛顿线性化方法。我们证明，在与相邻时间步长之比的温和限制下，提出的方案的无条件最佳误差估计值，即$ 0 <r_k <r _ {\ max} \ of 4.8645 $和最大时间步长。证明涉及离散的正交卷积（DOC）和离散互补卷积（DCC）内核以及误差分解方法。此外，我们的分析还表明，BDF1获得的第一级解决方案$ u^1 $（即向后的Euler方案）不会导致二阶全局准确性的损失。提供了数值示例以证明我们的理论结果。
In this paper we consider a linearized variable-time-step two-step backward differentiation formula (BDF2) scheme for solving nonlinear parabolic equations. The scheme is constructed by using the variable time-step BDF2 for the linear term and a Newton linearized method for the nonlinear term in time combining with a Galerkin finite element method (FEM) in space. We prove the unconditionally optimal error estimate of the proposed scheme under mild restrictions on the ratio of adjacent time-steps, i.e. $0<r_k < r_{\max} \approx 4.8645$ and on the maximum time step. The proof involves the discrete orthogonal convolution (DOC) and discrete complementary convolution (DCC) kernels, and the error splitting approach. In addition, our analysis also shows that the first level solution $u^1$ obtained by BDF1 (i.e. backward Euler scheme) does not cause the loss of global accuracy of second order. Numerical examples are provided to demonstrate our theoretical results.
