Numerical verification method for positive solutions of elliptic problems

椭圆方程解的正性数值验证方法

Numerical verification method for positiveness of solutions to elliptic equations

本文提出一种用于验证半线性椭圆方程解正性的数值方法，提供了一个可数值检查的充分条件，无需复杂计算，并通过实例展示。

A posteriori verification of the positivity of solutions to elliptic boundary value problems

The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither $H^2$-regularity nor $ L^{\infty} $-error estimation, but only $ H^1_0 $-error estimation. In [J. Comput. Appl. Math, Vol. 370, (2020) 112647], we proposed two approaches to verify the positivity of solutions of several semilinear elliptic boundary value problems. However, some cases require $ L^{\infty} $-error estimation and, therefore, narrow applicability. In this paper, we extend one of the approaches and combine it with a …

A posteriori verification for the sign-change structure of solutions of elliptic partial differential equations

This paper proposes a method for rigorously analyzing the sign-change structure of solutions of elliptic partial differential equations subject to one of the three types of homogeneous boundary conditions: Dirichlet, Neumann, and mixed. Given explicitly estimated error bounds between an exact solution $ u $ and a numerically computed approximate solution $ \hat{u} $, we evaluate the number of sign-changes of $ u $ (the number of nodal domains) and determine the location of zero level-sets of $ u $ (the location of the nodal line). We apply this method to the Dirichlet problem of the Allen-Cahn…

非协调有限元对半线性椭圆问题的保证inf-sup界与存在性验证

Guaranteed inf-sup bounds and existence verification for semilinear elliptic problems via nonconforming finite elements

本文提出一种基于牛顿-康德罗维奇论证的非协调有限元后验验证方法，利用新型误差界保证inf-sup下界，实现对半线性椭圆问题及二维稳态纳维-斯托克斯方程的解存在性验证。

A new formulation for the numerical proof of the existence of solutions to elliptic problems

Infinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of PDEs, the original problem is transformed into the infinite Newton-type fixed point equation $w = - {\mathcal L}^{-1} {\mathcal F}(\hat{u}) + {\mathcal L}^{-1} {\mathcal G}(w)$, where ${\mathcal L}$ is a linearized operator, ${\mathcal F}(\hat{u})$ is a residual, and ${\mathcal G}(w)$ is a local Lipschitz term. Therefore, the estimations of $\| {\mathcal L}^{-1} {\mathcal F}(\hat{u}) \|$ and $\| {\mathcal L…