数苑博雅讲座（66）：A unified analysis framework for high-order structure-preserving mimetic difference methods
来源：
发布时间：2026-05-29
报告时间：- 2026年6月2日（周二）上午09: 30-10: 30
- 报告地点：
- 国交2栋315会议室
- 报告人：
- 徐岩
- 报告人单位：
- 中国科学技术大学
- 邀请人：
- 王燕
- 摘要：
- We introduce a unified analytical framework for high-order numerical schemes based on mimetic discretization, applicable to a broad class of partial differential equations. The core of the framework lies in systematically characterizing the relationship between high-order and second-order mimetic operators. In particular, we show that high-order mimetic operators can be factorized into compositions of second-order mimetic operators and additional operators with simple structures. Within this setting, we design reduction operators that simultaneously control high-order local truncation errors while preserving the intrinsic geometric and physical structures of the underlying problems. A central feature of the framework is the construction of a mimetic de Rham complex that forms an exact sequence and commutes with the corresponding continuous de Rham complex. Building on this structure-preserving formulation, we establish high-order a priori error estimates for a variety of model problems. Furthermore, the proposed approach naturally enforces key physical constraints, including curl-free conditions in acoustic equations and divergence-free properties in Maxwell's equations. These results demonstrate the effectiveness of the framework in achieving both high-order accuracy and structure preservation. For each problem, we provide numerical experiments to verify both the designed accuracy and the preservation of these structures. These results demonstrate the effectiveness of the framework in achieving both high-order accuracy and structure preservation.
