奇异摄动半线性椭圆型方程多解计算方法及相关问题的研究
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摘要
本文研究了一类典型的奇异摄动Neumann问题,一类一般的奇异摄动Neumann问题以及一类半线性椭圆Dirichlet问题的多解计算理论与数值算法。对奇异摄动多解模型提出了改进型LMM算法并对算法的收敛性进行了分析,对半线性椭圆Dirichlet问题设计了基于Chebyshev谱配点法的SEM算法并对算法的收敛性进行了分析。本文还证明了奇异摄动多解模型的一些理论结果。
首先,本文通过放松传统局部极小极大算法(LMM)在选择初始上升方向时的严格正交性条件的限制,并引入局部加密网格策略及其它策略,提出了改进型LMM算法。证明了新算法产生的点列在满足峰选择p连续、可分离条件以及能量泛函下方有界三个条件之下一定能收敛到新解。本文的数值算例首次比较系统地计算得到了奇异摄动Neumann问题的最小能量解、边界角落处多峰解、边界非角落处多峰解、内部单峰解、内部多峰解以及其它一些特殊的解。
接下来,受数值结果启发,本文先后证明了奇异摄动模型(1-1)和(4-1)的平凡解在任意ε值时的Morse指标、平凡解的分叉点以及决定非平凡正解与非平凡负解是否存在的临界值εc1与εc2。讨论了最小能量解和变号解的一些性质,当ε充分大时,得到一个同类变号解的能量与最大值的增长阶分别为两p+1/p-1和1/p-1的猜想。这些理论结果被大量数值结果所验证。
最后,本文对一类半线性椭圆Dirichlet问题设计了基于Chebyshev谱配点法的SEM算法。在基本假定(5-18)下分析了该算法的收敛性,证明通过Chebyshev谱配点法求解模型方程(1-3)得到的数值解与其相对应的真解的Hω1和Hω2误差分别为O(N1-σ)和O(N-σ),它们反映了谱配点法的丰满阶。而且,当N充分大时,充分靠近某一真解u的数值解uN是唯一的。数值结果表明基于Chebyshev谱配点法的SEM算法比传统的基于有限元离散及两网格法的SEM算法效率更高。
In this paper, the theory and numerical methods for solving multiple solu-tions to a classical singularly perturbed Neumann problem, a general singularly perturbed Neumann problem and a semi-linear elliptic Dirichlet problem are investigated. A modified local minimax method (LMM) and its convergence analysis are proposed for the singularly perturbed multiple solution models. A search extension method (SEM) based on the Chebyshev spectral collocation method and its convergence analysis are displayed for the semi-linear ellip-tic Dirichlet problem. Some theoretical results for the singularly perturbed multiple solution models are also proved in this paper.
Firstly, by relaxing the strict orthogonality requirement in selecting an initial guess for the traditional local minimax method(LMM) and introducing the local refinement strategy and other strategies, a modified LMM algorithm is designed in this paper. The sequences generated by the new algorithm are proved to converge to a new solution if the three conditions that the peak selection p is continuous, the separable condition and the energy functional is bounded below are all satisfied. The least energy solutions, boundary cor-ner multi-peak solutions, boundary non-corner multi-peak solutions, interior single-peak solutions and interior multi-peak solutions of the singularly per-turbed Neumann problems are systematically computed for the first time in the numerical examples.
Next, motivated by the numerical results, the Morse indices of the trivial solutions at any value of ε, the bifurcation points of the trivial solutions and the critical values εc1and εc2, which determine the existence or nonexistence of a nontrivial positive solution and a nontrivial negative solution, are verified for the singularly perturbed models (1-1) and (4-1) respectively. The paper discusses some properties of the least energy solutions and sign-changing so-lutions and proposes a conjecture that the increasing orders of the energy and the maximum for the same kind of sign-changing solutions are p+1/p-1and1/p-1respectively, provided that ε is sufficiently large. All these theoretical results are proved by a large number of numerical results.
Finally, a SEM based on Chebyshev spectral collocation method is de-veloped for a semi-linear elliptic Dirichlet problem in this paper. Under the basic assumption (5-18), we analyze the convergence results of this method and show that the Hw1and Lw2errors between the numerical solutions by solv-ing model equation (1-3) with the Chebyshev spectral collocation method and the corresponding true solutions are O(N1-σ) and O(N-σ) respectively, which reflect the full orders of spectral collocation method. Furthermore, when N is sufficiently large, the numerical solution μN which sufficiently approaches a true solution u is unique. The numerical results state that the SEM based on Chebyshev spectral collocation method is more efficient than the traditional SEM based on finite element discretization and two-grid method.
首先,本文通过放松传统局部极小极大算法(LMM)在选择初始上升方向时的严格正交性条件的限制,并引入局部加密网格策略及其它策略,提出了改进型LMM算法。证明了新算法产生的点列在满足峰选择p连续、可分离条件以及能量泛函下方有界三个条件之下一定能收敛到新解。本文的数值算例首次比较系统地计算得到了奇异摄动Neumann问题的最小能量解、边界角落处多峰解、边界非角落处多峰解、内部单峰解、内部多峰解以及其它一些特殊的解。
接下来,受数值结果启发,本文先后证明了奇异摄动模型(1-1)和(4-1)的平凡解在任意ε值时的Morse指标、平凡解的分叉点以及决定非平凡正解与非平凡负解是否存在的临界值εc1与εc2。讨论了最小能量解和变号解的一些性质,当ε充分大时,得到一个同类变号解的能量与最大值的增长阶分别为两p+1/p-1和1/p-1的猜想。这些理论结果被大量数值结果所验证。
最后,本文对一类半线性椭圆Dirichlet问题设计了基于Chebyshev谱配点法的SEM算法。在基本假定(5-18)下分析了该算法的收敛性,证明通过Chebyshev谱配点法求解模型方程(1-3)得到的数值解与其相对应的真解的Hω1和Hω2误差分别为O(N1-σ)和O(N-σ),它们反映了谱配点法的丰满阶。而且,当N充分大时,充分靠近某一真解u的数值解uN是唯一的。数值结果表明基于Chebyshev谱配点法的SEM算法比传统的基于有限元离散及两网格法的SEM算法效率更高。
In this paper, the theory and numerical methods for solving multiple solu-tions to a classical singularly perturbed Neumann problem, a general singularly perturbed Neumann problem and a semi-linear elliptic Dirichlet problem are investigated. A modified local minimax method (LMM) and its convergence analysis are proposed for the singularly perturbed multiple solution models. A search extension method (SEM) based on the Chebyshev spectral collocation method and its convergence analysis are displayed for the semi-linear ellip-tic Dirichlet problem. Some theoretical results for the singularly perturbed multiple solution models are also proved in this paper.
Firstly, by relaxing the strict orthogonality requirement in selecting an initial guess for the traditional local minimax method(LMM) and introducing the local refinement strategy and other strategies, a modified LMM algorithm is designed in this paper. The sequences generated by the new algorithm are proved to converge to a new solution if the three conditions that the peak selection p is continuous, the separable condition and the energy functional is bounded below are all satisfied. The least energy solutions, boundary cor-ner multi-peak solutions, boundary non-corner multi-peak solutions, interior single-peak solutions and interior multi-peak solutions of the singularly per-turbed Neumann problems are systematically computed for the first time in the numerical examples.
Next, motivated by the numerical results, the Morse indices of the trivial solutions at any value of ε, the bifurcation points of the trivial solutions and the critical values εc1and εc2, which determine the existence or nonexistence of a nontrivial positive solution and a nontrivial negative solution, are verified for the singularly perturbed models (1-1) and (4-1) respectively. The paper discusses some properties of the least energy solutions and sign-changing so-lutions and proposes a conjecture that the increasing orders of the energy and the maximum for the same kind of sign-changing solutions are p+1/p-1and1/p-1respectively, provided that ε is sufficiently large. All these theoretical results are proved by a large number of numerical results.
Finally, a SEM based on Chebyshev spectral collocation method is de-veloped for a semi-linear elliptic Dirichlet problem in this paper. Under the basic assumption (5-18), we analyze the convergence results of this method and show that the Hw1and Lw2errors between the numerical solutions by solv-ing model equation (1-3) with the Chebyshev spectral collocation method and the corresponding true solutions are O(N1-σ) and O(N-σ) respectively, which reflect the full orders of spectral collocation method. Furthermore, when N is sufficiently large, the numerical solution μN which sufficiently approaches a true solution u is unique. The numerical results state that the SEM based on Chebyshev spectral collocation method is more efficient than the traditional SEM based on finite element discretization and two-grid method.
引文
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