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

该提案旨在为科学和工程偏微分方程（PDEs）解的新兴计算方法家族提供严格的数学背景。该方法是基于将原始的线性或非线性偏微分方程的边值问题简化为局部边界域积分或积分微分方程，这些方程经过基于网格或无网格的离散化，导致具有稀疏矩阵的代数方程系统。这对于具有可变系数的问题尤其有益，在这些问题中，没有可用的解析和/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来代替。变系数偏微分方程自然出现在非均匀线性和非线性介质的数学建模中（例如，功能梯度材料，具有损伤引起的不均匀性或弹性壳的材料），在固体力学，电磁学，导热性，流体通过多孔介质流动以及其他物理和工程领域。将偏微分方程的边值问题化为边界积分方程的关键是对原偏微分方程的基本解。但是，对于变系数的偏微分方程或模拟复杂介质的偏微分方程，通常没有分析和（或）廉价计算的形式。根据Levi和Hilbert的理论，在这种情况下，我们可以使用原始PDE的参数矩阵（Levi函数）作为基本解的替代品。参数化通常比基本解广泛得多，虽然不需要满足原PDE，但它正确地描述了基本解的主要部分。这使问题不再是边界积分方程，而是边界-域积分方程。它的离散化导致了一个与有限元法（FEM）相似大小的代数方程组，但是该系统的矩阵不像有限元法那样稀疏，因此对数值解的效率较低。用边界域积分方程法求解非线性问题（如非线性传热、弹性或大变形下的弹性壳）时也会出现类似的情况。最近出现的局部边界域积分方程方法解决了这一缺陷，使其与有限元法在此类问题上具有竞争力。它使用特殊构造的局部参数将线性和非线性变系数bvp简化为局部边界域积分或积分微分方程（LBDI(D)Es）。在局部支持的基于网格或无网格的离散化之后，这导致了计算效率高的代数方程的稀疏系统。LBDI(D) e的进一步发展，特别是探索它们可以通过迭代算法求解而不需要预处理的想法，由于它们具有良好的谱性质，需要对相应的积分算子和积分微分算子的性质有更深入的分析见解，而这正是该项目旨在提供的。该项目的分析结果将在PI指导下由两名博士生开发的数值算法和计算机代码中实施，这两名博士生由其他来源支持，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.