Mathematical analysis of Localised Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems

该提议旨在为科学和工程中的偏微分方程组(PDE)的求解发展一种新的计算方法家族的严格的数学背景。该方法基于将偏微分方程组的线性或非线性边值问题归结为局部边界域积分或积分-微分方程组，经过网格离散或无网格离散后得到具有稀疏矩阵的代数方程组。这对于具有可变系数的问题尤其有益，其中没有解析和/或廉价计算形式的基本解，但该方法采用了广泛可用的局部参数线。变系数偏微分方程组在固体力学、电磁学、导热学、流体流动等物理和工程领域的非均匀线性和非线性介质(如功能梯度材料、损伤引起的非均匀材料或弹性壳)的数学模拟中自然产生。将偏微分方程边值问题化为边界积分方程解的主要内容是原始偏微分方程解的基本解。然而，对于具有可变系数的偏微分方程组或模拟复杂介质的偏微分方程组，它通常不能以解析和/或廉价计算的形式提供。在李维和希尔伯特之后，在这种情况下，人们可以使用原始偏微分方程组的参数线(李维函数)来代替基本解。参数矩阵的适用范围通常比基本解广泛得多，并正确地描述了基本解的主要部分，尽管不一定要满足原始的偏微分方程。这使得问题不再是边界积分方程解，而是边界域积分方程解。它的离散化导致了一个与有限元方法相似大小的代数方程组，但系统的矩阵不像有限元方法那样稀疏，从而降低了数值求解的效率。当用边界域积分方程法求解非线性问题(如非线性热传导、弹性壳或大变形弹性壳)时，也会出现类似的情况。局部边界域积分方程法是最近出现的一种方法，它克服了这一缺陷，使其在这类问题上与有限元方法相竞争。它利用特殊构造的局部参数将变系数的线性和非线性边值问题化为局部边界域积分或积分-微分方程组，LBDI(D)es。在局部支持的基于网格或无网格的离散化之后，这将导致稀疏的代数方程组对计算有效。LBDI(D)方程的进一步发展，特别是探索由于其良好的谱性质，可以通过不需要预条件的迭代算法来求解的思想，需要对相应的积分和积分-微分算子的性质有更深入的分析见解，本项目的目的是提供这些性质。项目分析结果将由两名博士生在PI监督下开发的数值算法和计算机代码中实施，这两名博士生得到了其他来源的支持，因此不包括在提案中。

The proposal is aimed at developing a rigorous mathematical backgrounds of an emerging new family of computational methods for solution of partial differential equations (PDEs) of science and engineering. The approach is based on reducing the original linear or nonlinear boundary value problems for PDEs to localised boundary-domain integral or integro-differential equations, which after mesh-based or mesh-less discretisation lead to systems of algebraic equations with sparse matrices. This is especially beneficial for problems with variable coefficients, where no fundamental solution is available in an analytical and/or cheaply calculated form, but the approach employs a widely available localised parametrix instead. PDEs with variable coefficients arise naturally in mathematical modelling non-homogeneous linear and nonlinear media (e.g. functionally graded materials, materials with damage-induced inhomogeneity or elastic shells) in solid mechanics, electromagnetics, thermo-conductivity, fluid flows trough porous media, and other areas of physics and engineering. The main ingredient for reducing a boundary-value problem for a PDE to a boundary integral equation is a fundamental solution to the original PDE. However, it is generally not available in an analytical and/or cheaply calculated form for PDEs with variable coefficients or PDEs modelling complex media. Following Levi and Hilbert, one can use in this case a parametrix (Levi function) to the original PDE as a substitute for the fundamental solution. Parametrix is usually much wider available than fundamental solution and correctly describes the main part of the fundamental solution although does not have to satisfy the original PDE. This reduces the problem not to boundary integral equation but to boundary-domain integral equation. Its discretisation leads to a system of algebraic equations of the similar size as in the finite element method (FEM), however the matrix of the system is not sparse as in the FEM and thus less efficient for numerical solution. Similar situation occurs also when solving nonlinear problems (e.g. for non-linear heat transfer, elasticity or elastic shells under large deformations) by boundary-domain integral equation method. The Localised Boundary-Domain Integral Equation method emerged recently addressing this deficiency and making it competitive with the FEM for such problems. It employs specially constructed localised parametrices to reduce linear and non-linear BVPs with variable coefficients to Localised Boundary-Domain Integral or Integro-Differential Equations, LBDI(D)Es. After a locally-supported mesh-based or mesh-less discretisation this leads to sparse systems of algebraic equations efficient for computations. Further development of the LBDI(D)Es, particularly exploring the idea that they can be solved by iterative algorithms needing no preconditioning, due to their favourable spectral properties, requires a deeper analytical insight into properties of the corresponding integral and integro-differential operators, which the project is aimed to provide. The project analytical results will be implemented in numerical algorithms and computer codes developed under the PI supervision by two PhD students, who are supported from other sources and thus are not included the proposal.