Low-dimensional qualitative and numerical approximation of bifurcation in a semilinear elliptic problem
In this interesting paper, bifurcation phenomena are investigated for the following nonlinear boundary value problem, treated as a model for more complicated situations: \(u_{xx}+u_{yy}+g(u)=0,\) (x,y)\(\in \Omega\); \(u(x,y)=0\), (x,y)\(\in \partial \Omega\). Here the function g is of some special form and may depend on parameters. (One parameter is the dimension of the rectangle \(\Omega)\). The considered problem admits always the trivial solution but also some nonlocal bifurcations (non- trivial solutions, appearing spontaneously for certain values of the parameters) are theoretically expected. Developing the solution as double Fourier series with respect to the eigenfunctions of the Laplace operator on the rectangle \(\Omega\), a reduction process is performed. This process reduces the original problem to some finite-dimensional one. First, the author investigates a Galerkin approximation of the reduced problem, taking into account a subspace generated by only one eigenfunction. Then the same problem is treated numerically, and the results are compared. The paper contains a rather detailed description of the algorithm applied to the numerical solution of a nonlinear parameter- dependent system of equations, resulting after the reduction process. Some results of computations are presented.
scientific article; zbMATH DE number 4058788