摘 要：偏微分方程数值解法的收敛性分析是计算数学领域的重要研究课题，随着科学与工程问题对高精度数值模拟需求的增长，深入探讨数值方法的收敛特性具有重要意义。本文聚焦于有限差分法和有限元法在求解椭圆型、抛物型及双曲型偏微分方程时的收敛性，通过引入新的离散能量估计技术，建立了统一的理论框架。研究采用严格的数学推导，结合算子半群理论与Sobolev空间理论，证明了在不同边界条件下数值解的一致收敛性，并给出了最优误差阶估计。特别地，针对非线性问题提出了改进的迭代格式，有效提高了算法稳定性。数值实验验证了理论结果的正确性，表明所提方法在处理复杂几何区域和奇异解方面具有显著优势。本研究不仅丰富了偏微分方程数值解法的理论体系，还为实际应用提供了可靠的理论依据和技术支持。
关键词：偏微分方程数值解；收敛性分析；有限差分法
Abstract：The convergence analysis of numerical methods for partial differential equations (PDEs) is a crucial research topic in computational mathematics. As the demand for high-precision numerical simulations in scientific and engineering problems increases, an in-depth investigation into the convergence properties of numerical methods becomes increasingly significant. This study focuses on the convergence of finite difference methods and finite element methods when solving elliptic, parabolic, and hyperbolic PDEs. By introducing novel discrete energy estimation techniques, a unified theoretical fr amework has been established. The research employs rigorous mathematical derivations, combining semigroup theory of operators and Sobolev space theory, to prove uniform convergence of numerical solutions under various boundary conditions and provides optimal error estimates. Specifically, an improved iterative scheme for nonlinear problems has been proposed, effectively enhancing algorithm stability. Numerical experiments validate the correctness of the theoretical results, demonstrating that the proposed methods exhibit significant advantages in handling complex geometrical domains and singular solutions. This study not only enriches the theoretical system of numerical methods for PDEs but also provides reliable theoretical support and technical guidance for practical applications.
引言 1
一、收敛性理论基础 1
(一)偏微分方程数值解概述 1
(二)收敛性定义与分类 2
(三)误差分析基本概念 2
二、离散格式的收敛性 3
(一)有限差分法收敛性 3
(二)有限元法收敛特性 3
(三)有限体积法收敛分析 4
三、时间离散化方法 4
(一)显式时间步进法 4
(二)隐式时间步进法 5
(三)时间耦合收敛性 5
四、收敛性加速技术 6
(一)多重网格方法应用 6
(二)并行计算对收敛性影响 6
(三)自适应网格加密策略 7
结论 7
参考文献 9
致谢 9
关键词：偏微分方程数值解；收敛性分析；有限差分法
Abstract：The convergence analysis of numerical methods for partial differential equations (PDEs) is a crucial research topic in computational mathematics. As the demand for high-precision numerical simulations in scientific and engineering problems increases, an in-depth investigation into the convergence properties of numerical methods becomes increasingly significant. This study focuses on the convergence of finite difference methods and finite element methods when solving elliptic, parabolic, and hyperbolic PDEs. By introducing novel discrete energy estimation techniques, a unified theoretical fr amework has been established. The research employs rigorous mathematical derivations, combining semigroup theory of operators and Sobolev space theory, to prove uniform convergence of numerical solutions under various boundary conditions and provides optimal error estimates. Specifically, an improved iterative scheme for nonlinear problems has been proposed, effectively enhancing algorithm stability. Numerical experiments validate the correctness of the theoretical results, demonstrating that the proposed methods exhibit significant advantages in handling complex geometrical domains and singular solutions. This study not only enriches the theoretical system of numerical methods for PDEs but also provides reliable theoretical support and technical guidance for practical applications.
Keywords: Numerical Solution Of Partial Differential Equations；Convergence Analysis；Finite Difference Method
引言 1
一、收敛性理论基础 1
(一)偏微分方程数值解概述 1
(二)收敛性定义与分类 2
(三)误差分析基本概念 2
二、离散格式的收敛性 3
(一)有限差分法收敛性 3
(二)有限元法收敛特性 3
(三)有限体积法收敛分析 4
三、时间离散化方法 4
(一)显式时间步进法 4
(二)隐式时间步进法 5
(三)时间耦合收敛性 5
四、收敛性加速技术 6
(一)多重网格方法应用 6
(二)并行计算对收敛性影响 6
(三)自适应网格加密策略 7
结论 7
参考文献 9
致谢 9
