4 research outputs found
Convergence analysis of a spectral-Galerkin-type search extension method for finding multiple solutions to semilinear problems
In this paper, we develop an efficient spectral-Galerkin-type search
extension method (SGSEM) for finding multiple solutions to semilinear elliptic
boundary value problems. This method constructs effective initial data for
multiple solutions based on the linear combinations of some eigenfunctions of
the corresponding linear eigenvalue problem, and thus takes full advantage of
the traditional search extension method in constructing initials for multiple
solutions. Meanwhile, it possesses a low computational cost and high accuracy
due to the employment of an interpolated coefficient Legendre-Galerkin spectral
discretization. By applying the Schauder's fixed point theorem and other
technical strategies, the existence and spectral convergence of the numerical
solution corresponding to a specified true solution are rigorously proved. In
addition, the uniqueness of the numerical solution in a sufficiently small
neighborhood of each specified true solution is strictly verified. Numerical
results demonstrate the feasibility and efficiency of our algorithm and present
different types of multiple solutions.Comment: 23 pages, 7 figures; Chinese version of this paper is published in
SCIENTIA SINICA Mathematica, Vol. 51 (2021), pp. 1407-143
Adaptive local minimax Galerkin methods for variational problems
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