积分-微分方程是广泛应用于物理、化学和生物等领域的一类重要数学模型，研究其高性能数值解法具有重要的理论意义和应用价值。本项目拟研究Fredholm积分-微分方程的两种多尺度快速算法：(1)针对大规模、高精度问题在求解离散化所得方程组遇到的存储量大、条件数高和计算时间长的瓶颈问题，我们拟采用Sobolev空间上的多尺度正交基来离散化方程，以优化方程组的代数结构，并在此基础上设计快速求解离散化方程组的多层扩充法。预期算法以准线性计算复杂度得到最优收敛阶。(2)考虑到积分项在离散化过程中占据大量的计算时间，我们将采用L^2空间上的多尺度正交函数作基底，利用基底的消失矩性和积分算子的紧性所带来的离散化系数矩阵的“数值稀疏性”，给出截断策略，实现方程的快速离散化并获得系数矩阵稀疏的代数方程组，发展求解该方程的多尺度间断Galerkin快速算法。研究成果将进一步丰和发展多尺度数值计算方法的理论。

Integro-differential equation serves as an important mathematical model in the study of physical, chemical and biological phenomena. Developping efficient numerical methods for solving this kind of equations has important theoretical significance and great value in application. In this research, the following two kinds of fast multiscale algorithms for Fredholm integro-differential equations will be studied: (1) For large-scale and high accuracy problems, it requires us to use sufficiently fine grids which demands a large amount of storage space and computational effort, and the condition number of the coefficient matrices are possibly large. To overcome this bottleneck, we will use the multiscale orthonormal bases in Sobolev space to discrete the integro-differential equation which can optimize the algebraic structure of the resulting linear systems, then we will develop a multilevel augmentation method for fast solving the linear systems. (2) Considering that the integral term occupies a large amount of computational time in the discretization process, we will choose multiscale orthonormal functions in L^2 space as the basis. Because the vanishing moments of this bases and the compactness of the integral operator lead to “numerical sparsity” of the resulting matrix, we will propose a truncation strategy to fast discrete the equation and obtain resulting equations with sparse coefficient matrix. Then, a multiscale discontinuous Galerkin method will be developed. The research of this project will further enrich the computational theory of the multiscale methods.