Some Second-Order <i>σ</i> Schemes Combined with an <i>H</i>¹-Galerkin MFE Method for a Nonlinear Distributed-Order Sub-Diffusion Equation

• Main Authors: Yaxin Hou, Cao Wen, Hong Li, Yang Liu, Zhichao Fang, Yining Yang
• Format: Article
• Language: English
• Published: MDPI AG 2020-02-01
• Series: Mathematics
• Subjects: second-order <i>σ</i> scheme; interpolation approximation; <i>h</i>¹-galerkin mixed finite element method; nonlinear distributed-order sub-diffusion equation; stability; error estimates
• Online Access: https://www.mdpi.com/2227-7390/8/2/187
• ISSN: 2227-7390

In this article, some high-order time discrete schemes with an <inline-formula> <math display="inline"> <semantics> <msup> <mi>H</mi> <mn>1</mn> </msup> </semantics> </math> </inline-formula>-Galerkin mixed finite element (MFE) method are studied to numerically solve a nonlinear distributed-order sub-diffusion model. Among the considered techniques, the interpolation approximation combined with second-order <inline-formula> <math display="inline"> <semantics> <mi>&#963;</mi> </semantics> </math> </inline-formula> schemes in time is used to approximate the distributed order derivative. The stability and convergence of the scheme are discussed. Some numerical examples are provided to indicate the feasibility and efficiency of our schemes.