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

We study the numerical computation of nontrivial (typically saddle-type) critical points of variational functionals associated with nonlinear Dirichlet problems involving the $p$-Laplacian. While classical numerical mountain pass algorithms rely on path deformations or finite element discretizations, we propose a different approach based on a geometric B-spline representation of the solution. The idea is to parameterize the function space by smooth spline curves and perform a mountain-pass type \emph{up--down} iteration directly in the space of control points. This transforms the infinite-dimensional mountain pass problem into a finite-dimensional geometric problem and introduces tools from spline theory into nonlinear elliptic PDE computation. The descent direction is obtained through an auxiliary Poisson equation, yielding a Sobolev gradient that significantly stabilizes the iteration. Convergence is monitored via both the gradient norm and the Euler--Lagrange residual, ensuring that the resulting B-spline approximation satisfies the underlying PDE. Numerical experiments for the model case p = 2 and f(u) = u ³ on Ω =(0, 1) demonstrate that the method computes nontrivial solutions consistent with mountain-pass geometry; depending on initialization, the computed profiles may include sign-changing solutions. The results highlight the effectiveness and flexibility of B-spline geometry in nonlinear variational problems and suggest new directions for numerical mountain pass methods based on geometric parametrizations.