Monge–Ampére simulation of fourth order PDEs in two dimensions with application to elastic–electrostatic contact problems

Constructions of the soliton solutions to coupled nonlinear Schrödinger equation with advanced mathematical techniques

“…Various numerical techniques are frequently employed for this purpose, including the Galerkin finite element method (GFEM), 29 the finite difference method (FDM), [30][31][32][33] the compact finite-difference method, 34 and the adaptive mesh refinement method. [35][36][37][38][39][40] The adaptive mesh refinement method, in particular, stands out as a potent approach for efficiently tackling PDEs with a high degree of accuracy, making it especially valuable for simulating intricate and dynamic phenomena. For nonlinear PDEs, iteration techniques like the Newton-Raphson method, Picard iteration, or fixed-point iteration are commonly used to handle nonlinear terms.…”

A Study of Traveling Wave Structures and Numerical Investigations into the Coupled Nonlinear Schrödinger Equation Using Advanced Mathematical Techniques

“…Furthermore, numerical methods are often required for solving nonlinear PDEs when analytical solutions are not feasible. Some commonly used numerical methods to solve nonlinear PDEs include the finite difference method (FDM) [8,9], the compact finite-difference method [10], the Galerkin finite element method (GFEM) [11], and the adaptive mesh refinement method, which is a powerful approach for efficiently solving PDEs while maintaining high accuracy, making it particularly valuable for simulations involving complex and dynamic phenomena [12][13][14][15][16], and more others [17][18][19]. The chosen method depends on the problem's nature, the desired accuracy, computational resources, and available software libraries.…”

Moving mesh simulation of contact sets in two dimensional models of elastic–electrostatic deflection problems

“…In previous computational studies of singularity formation in second-order PDEs, moving mesh methods based on parabolic Monge-Ampére (PMA) discretization have been successfully employed in one-dimensional [8] or rectangular two-dimensional domains [6,7,9]. The PMA moving mesh approach has recently [10] been extended to the fourth-order PDE problem (1) by constructing a high regularity mapping between the computational and physical domains. The study [10] was based on a finite difference discretization of the PMA equation that restricted computations to rectangular domains.…”

“…The PMA moving mesh approach has recently [10] been extended to the fourth-order PDE problem (1) by constructing a high regularity mapping between the computational and physical domains. The study [10] was based on a finite difference discretization of the PMA equation that restricted computations to rectangular domains. The main contribution of the present work is a robust adaptive numerical method that can resolve the singularities of (1) in general non-simply connected two-dimensional regions such as those utilized in real MEMS devices (cf.…”