Parametric nonlinear PDEs with multiple solutions: A PGD approach

This paper presents some insights into the determination, using the Proper Generalized Decomposition, of multiple solutions of nonlinear parametric partial differential equations. Although the Proper Generalized Decomposition (PGD) is well suited for computing the solution of, possibly nonlinear, parametric problems that vary smoothly with a physical parameter, no work has been achieved for the case of problems that exhibit multiple solutions for some values of a parameter. For two representative cases, we show how an appropriate parametrization, combined to a nonlinear solution procedure can be devised to describe and compute the multiple solutions of a PDE.