On a nonlinear problem with Dirichlet and Acoustic boundary conditions

摘要

The aims of this paper are to establish theoretical analysis and numerical simulation for a nonlinear wave equation with mixed boundary conditions of Dirichlet and Acoustic type. The theoretical results are about: existence and uniqueness of global solutions, regularity and uniform stability of these global solutions and an exponential decay rate for energy. In the numerical context, simulations are presented using the finite element method in space (with linear and quadratic Lagrange basis), the Crank-Nicolson method in time and, for each discrete time, the Newton’s method is used to solve the nonlinear algebraic system. Furthermore, the energy exponential decay and convergence order (sub-optimal and optimal) are presented numerically.