an:05837466
Zbl 1203.65118
Shen, Ting-Ting; Zhang, Zhong-Qiang; Ma, He-Ping
Optimal error estimates of the Legendre tau method for second-order differential equations
EN
J. Sci. Comput. 42, No. 2, 198-215 (2010).
00272482
2010
j
65L60 65L70 65M15 65M70
tau method; optimal error estimate; second-order differential equation
Summary: We prove that the Legendre tau method has the optimal rate of convergence in \(L ^{2}\)-norm, \(H ^{1}\)-norm and \(H ^{2}\)-norm for one-dimensional second-order steady differential equations with three kinds of boundary conditions and in \(C([0,T];L ^{2}(I))\)-norm for the corresponding evolution equation with the Dirichlet boundary condition. For the generalized Burgers equation, we develop a Legendre tau-Chebyshev collocation method, which can also be optimally convergent in \(C([0,T];L ^{2}(I))\)-norm. Finally, we give some numerical examples.