SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1108-A1133, January 2021.
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 nonstandard, 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.

中文翻译：

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变分问题的自适应局部极大极小Galerkin方法

SIAM科学计算杂志，第43卷，第2期，第A1108-A1133页，2021年1月。
在许多实际应用中，偏微分方程模型的解决方案作为基础（能量）函数的临界点出现。如果这样的解决方案是鞍点，而不是最大值或最小值，则理论框架将是非标准的，因此开发合适的数值逼近程序将面临巨大挑战。在本文中，我们的目的是为具有多个（鞍点）解的非线性变分问题的数值解提供一种迭代离散化方法。与通常在这种情况下通常失败的传统数值逼近方案相反，当前工作的关键思想是采用先前开发的局部极小极大值方法和自适应Galerkin离散化的同时相互作用。因此，我们得出了一种自适应局部极小极大伽勒金（LMMG）方法，该方法以高效的方式结合了对鞍点解及其在有限维空间中的逼近的搜索。在某些假设下，我们将证明生成的近似解序列收敛于变分问题的解集。该通用框架将应用于（奇摄动）半线性椭圆边值问题的有限元离散化的特定环境，并将进行一系列数值实验。我们将证明生成的近似解序列收敛于变分问题的解集。该通用框架将应用于（奇摄动）半线性椭圆边值问题的有限元离散化的特定环境，并将进行一系列数值实验。我们将证明生成的近似解序列收敛于变分问题的解集。该通用框架将应用于（奇摄动）半线性椭圆边值问题的有限元离散化的特定环境，并将进行一系列数值实验。