两类高阶偏微分方程的高效数值算法研究

高阶偏微分方程在材料科学、电磁学、流体力学、随机过程等领域具有广泛的应用。其中双调和方程、相场方程和非散度型椭圆方程在实际中越来越受到重视，因此研究上述方程的高效数值算法极为重要。本项目内容主要包括：.(1). 基于“离散导数”，构造双调和方程的非协调元算法并建立算法的理论分析框架；.(2). 应用上述新型算法，构造并分析相场方程和非散度型椭圆方程的高效数值算法。.本项目建立在申报人长期的工作积累和对国内外研究趋势的深刻理解之上，有望实现若干具有应用背景的高阶偏微分方程的大规模数值模拟。项目结果将推动非协调元方法的发展并辅助实际工程问题求解。

Higher-order partial differential equations have a wide range of applications in materials science, electromagnetics, hydrodynamics and stochastic processes. In practice, biharmonic equations, phase field equations and non-divergence elliptic equations have received more and more attention . Therefore, it is of great importance to investigate efficient numerical algorithms for the above high-order PDEs. In this project, we will consider the following works:.(1). Based on the "discrete derivative", the theoretical framework of the nonconforming finite element methods for biharmonic equations will be established;.(2). Applying the above new numerical schemes, we prefer to construct and analyze efficient numerical algorithms for phase-field equations and non-divergence elliptic equations;.This project is based on the applicant's long-term work accumulation and deep understanding of research trends at home and abroad. It is expected to accomplish large-scale numerical simulations of several high-order partial differential equations with application background. The results of the project will promote the development of the nonconforming finite element methods and assist in solving practical engineering problems.