论文标题
自适应局部最小盖尔金方法用于变异问题
Adaptive local minimax Galerkin methods for variational problems
论文作者
论文摘要
在许多实践利益的应用中，部分微分方程模型的解决方案是基础（能量）功能的关键点。如果这种解决方案是马鞍点，而不是最大值或最小值，那么理论框架是非标准的，并且开发合适的数值近似程序是高度挑战性的。在本文中，我们的目的是为具有多个（鞍点）解决方案的非线性变分问题的数值解提供一种迭代离散方法。与通常在这种情况下失败的传统数值近似方案相反，当前工作的关键思想是采用先前开发的局部最小值方法和自适应Galerkin离散的同时相互作用。因此，我们得出了一种自适应的局部最小盖尔金（LMMG）方法，该方法将搜索鞍点解的搜索及其在有限维空间中以非常有效的方式结合在一起。在某些假设下，我们将证明生成的近似解决方案的生成序列会收敛到变分问题的解决方案集。该一般框架将应用于（奇异）半连续性椭圆边界值问题的有限元离散的特定上下文，并将提出一系列的数值实验。
In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the theoretical framework is non-standard, and the development of suitable numerical approximation procedures turns out to be highly challenging. In this paper, our aim is to present an iterative discretization methodology for the numerical solution of nonlinear variational problems with multiple (saddle point) solutions. In contrast to traditional numerical approximation schemes, which typically fail in such situations, the key idea of the current work is to employ a simultaneous interplay of a previously developed local minimax approach and adaptive Galerkin discretizations. We thereby derive an adaptive local minimax Galerkin (LMMG) method, which combines the search for saddle point solutions and their approximation in finite-dimensional spaces in a highly effective way. Under certain assumptions, we will prove that the generated sequence of approximate solutions converges to the solution set of the variational problem. This general framework will be applied to the specific context of finite element discretizations of (singularly perturbed) semilinear elliptic boundary value problems, and a series of numerical experiments will be presented.
