Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs

该提议旨在为解非线性偏微分方程组(PDE)的新兴计算方法家族发展严格的数学背景。该方法将偏微分方程组的非线性边值问题归结为整体或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非线性代数方程组。在局域BDI(D)E的情况下，相应的代数方程的矩阵将是稀疏的。非线性偏微分方程组是在非线性物理过程的数学模型中自然产生的，如导热系数依赖于点温度和坐标的材料中的非线性热传导、损伤引起的非均匀材料、弹塑性材料、定常势可压缩流动的非线性方程、多孔介质中的非线性流动、非线性电磁学等物理和工程领域。将线性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偏微分方程解的基本解。然而，对于具有可变系数的线性偏微分方程组和非线性偏微分方程组，它通常不能以解析和/或廉价计算的形式获得。发展了Levi和Hilbert的思想，在这种情况下，人们可以使用参数线(Levi函数)来代替原始的非线性偏微分方程组或另一个线性偏微分方程组作为基本解的替代。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。这通常将非线性边值问题归结为一个整体非线性边值问题，而不是一个边界积分方程解。对一个整体非线性BDIDE系统进行离散化，得到一个与有限元方法相似大小的非线性代数方程组，但系统的矩阵不是稀疏的。非线性问题的局部化边界域积分-微分方程组(LBDIDES)最近出现，弥补了这一不足，使其在这类问题上与有限元方法相竞争。LBDIDE方法使用特殊构造的局部化参数将变系数的非线性边值问题化为LBDIDE。在采用局部支持的基于网格或无网格的离散化后，这导致了稀疏的非线性代数方程组的计算效率。然而，这一思想的实现需要对相应的非线性积分和积分-微分算子的性质有更深入的分析。申请人出版物中提供了线性情况下的全局和局部BDIE以及一些全球间接非线性BDIE的分析。该项目的目的是从这些结果跃升到对更一般的非线性、全局和局部BDIDE的分析。该项目的进一步发展涉及到求解全局或局部非线性BDIDE的迭代算法，特别是基于不动点定理。预计该项目的分析结果将在博士生的PI指导下开发的数值算法和计算机代码中实施。

The proposal is aimed at developing rigorous mathematical backgrounds of an emerging new family of computational methods for solution of nonlinear Partial Differential Equations (PDEs). The approach is based on reducing the original nonlinear boundary value problems for PDEs to global or localised Boundary-Domain Integral or Integro-Differential Equations, BDI(D)Es, which after mesh-based or mesh-less discretisation lead to nonlinear systems of algebraic equations. In case of localised BDI(D)Es, the matrices of corresponding algebraic equations will be sparse. Nonlinear PDEs arise naturally in mathematical modelling of nonlinear physical processes, e.g. of nonlinear heat transfer in materials with the thermo-conductivity coefficients depending on the point temperature and coordinate, materials with damage-induced inhomogeneity, elasto-plastic materials, nonlinear equation of stationary potential compressible flow, nonlinear flows trough porous media, nonlinear electromagnetics and other areas of physics and engineering. The main ingredient for reducing a boundary-value problem for a linear 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 linear PDEs with variable coefficients and for nonlinear PDEs. Developing ideas of Levi and Hilbert, one can use in this case a parametrix (Levi function) either to the original nonlinear PDE or to another, linear, 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 generally reduces the nonlinear boundary value problem not to a boundary integral equation but to a global nonlinear boundary-domain integro-differential equation. A discretisation of a global nonlinear BDIDE system leads to a system of nonlinear algebraic equations of the similar size as in the finite element method (FEM), however the matrix of the system is not sparse. The Localised Boundary-Domain Integro-Differential Equations, LBDIDEs, for nonlinear problems, emerged recently addressing this deficiency and making them competitive with the FEM for such problems. The LBDIDE method employs specially constructed localised parametrices to reduce nonlinear BVPs with variable coefficients to LBDIDEs. After employing a locally supported mesh-based or mesh-less discretisation, this leads to sparse systems of nonlinear algebraic equations efficient for computations. However implementation of this idea requires a deeper analytical insight into properties of the corresponding nonlinear integral and integro-differential operators. Such analysis is available in the applicants publications for the global and localised BDIEs in the linear case, and for some global indirect non-linear BDIEs. The project is intended to make a leap from these results to the analysis of much more general nonlinear global and localised BDIDEs. Further development of the project concerns the iterative algorithms to solve the global or localised nonlinear BDIDEs, particularly based on the fixed-point theorems. It is also expected that the project analytical results will be implemented in numerical algorithms and computer codes developed under the PI supervision by PhD students.