We study the numerical computation of nontrivial critical points of variational functionals associated with nonlinear Dirichlet problems involving the p-Laplacian. Previous numerical mountain pass approaches typically relied on finite element discretizations of the underlying function space. In contrast, we employ a discretization based on a geometric B-spline representation of the solution. The function space is approximated by smooth spline curves parameterized by control points, yielding a finite-dimensional geometric representation of the variational problem. Within this discrete space we apply a mountain-pass type up–down method. This allows the search for saddle-type critical points to be carried out directly in the space of spline control points. The descent direction is obtained through an auxiliary Poisson equation, providing a Sobolev gradient that stabilizes the iteration. Convergence of the numerical procedure is monitored via the Euler–Lagrange residual, ensuring that the computed spline approximation satisfies the variational problem up to a prescribed tolerance. Numerical experiments for the model case <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> with nonlinearity <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f(u)=u^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>u</mi> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega =(0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> show that the method computes nontrivial solutions, including sign-changing profiles depending on the initialization. The results demonstrate the successful application of B-splines in the numerical solution of nonlinear variational problems.

A Geometric B-Spline Approach to Mountain-Pass Type Solutions of Nonlinear Dirichlet Problems

期刊论文
出版模式：

开放获取
发表日期：

2026年4月13日

查看全文

Boróka Olteán-Péter

摘要

We study the numerical computation of nontrivial critical points of variational functionals associated with nonlinear Dirichlet problems involving the p-Laplacian. Previous numerical mountain pass approaches typically relied on finite element discretizations of the underlying function space. In contrast, we employ a discretization based on a geometric B-spline representation of the solution. The function space is approximated by smooth spline curves parameterized by control points, yielding a finite-dimensional geometric representation of the variational problem. Within this discrete space we apply a mountain-pass type up–down method. This allows the search for saddle-type critical points to be carried out directly in the space of spline control points. The descent direction is obtained through an auxiliary Poisson equation, providing a Sobolev gradient that stabilizes the iteration. Convergence of the numerical procedure is monitored via the Euler–Lagrange residual, ensuring that the computed spline approximation satisfies the variational problem up to a prescribed tolerance. Numerical experiments for the model case $p=2$ $p = 2$ with nonlinearity $f(u)=u^3$ $f (u) = u^{3}$ on $\Omega =(0,1)$ $Ω = (0, 1)$ show that the method computes nontrivial solutions, including sign-changing profiles depending on the initialization. The results demonstrate the successful application of B-splines in the numerical solution of nonlinear variational problems.