Discrete and continuous maximum principles for parabolic and elliptic operators
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摘要
The major qualitative properties of linear parabolic and elliptic operators/PDEs are the different maximum principles (MPs). Another important property is the stabilization property (SP), which connects these two types of operators/PDEs. This means that under some assumptions the solution of the parabolic PDE tends to an equilibrium state when t→∞, which is the solution of the corresponding elliptic PDE. To solve PDEs we need to use some numerical methods, and it is a natural requirement that these qualitative properties are preserved on the discrete level. In this work we investigate this question when a two-level discrete mesh operator is used as the discrete model of the parabolic operator (which is a one-step numerical procedure for solving the parabolic PDE) and a matrix as a discrete elliptic operator (which is a linear algebraic system of equations for solving the elliptic PDE). We clarify the relation between the discrete parabolic maximum principle (DPMP), the discrete elliptic maximum principle (DEMP) and the discrete stabilization property (DSP). The main result is that the DPMP implies the DSP and the DEMP.
论文关键词:Maximum principle,Discrete maximum principle,Stabilization,Matrix splitting,One-step iteration
论文评审过程:Received 31 October 2009, Revised 12 May 2010, Available online 8 June 2010.
论文官网地址:https://doi.org/10.1016/j.cam.2010.05.050