Homotopy Methods For Finding Multiple Solutions Of Elliptic Equations With Polynomial Nonlinearity

Semilinear elliptic equations or elliptic systems of equations arise in many models in science and engineering. The main focus of this dissertation is on numerically finding mul-tiple solutions of elliptic equations with polynomial nonlinearity. It contains the following aspects.1. Eigenfunction expansion discretization (EED) is considered for finding multiple solu-tions of semilinear elliptic equations with polynomial nonlinearity. Error estimates of EED, in H1-norm and L2-norm respectively, are derived. EED is proved to possess spectral accu-racy. Based on the error estimates, a filter strategy on successively finer levels is designed to remove spurious solutions and to refine nonspurious solutions, simultaneously. The filtered solutions are further refined by a finite-element-Newton method.2. For the polynomial systems resulted from EED, the standard homotopy seems not efficient for computing all solutions of them, when the number of eigenfunctions in EED is increased. For all solutions of the polynomial systems, an efficient extension homotopy is constructed, which is based on the structure of the polynomial systems and is composed of deformations on successively finer levels. When all solutions of the polynomial system on a level are obtained, all solutions of the polynomial system on the finer level can be efficiently computed. Smoothness and accessability of the homopoty paths are analyzed.3. A conjecture made by Chen and Xie is proved, which states that in eigenfunction expansion discretization for-△u=u3, when the finite dimensional subspace is taken as the eigensubspace corresponding to an N-fold eigenvalue of-△, the discretized problem has at least3N—1distinct nonzero real solutions. In the proof,"at least3N—1" is sharpened to "exactly3N—1". The analogies of this conjecture to the cases of cubic nonlinearity in three dimensions and quintic nonlinearity in two dimensions are considered and some key results are obtained. Also, two related results on the multiplicities of eigenvalues of—△in two dimensions and in three dimensions are presented.4. For elliptic equations with polynomial nonlinearity in the unit square, the solution set of the polynomial system resulted from EED is proved to inherit the symmetry of the solution set of the boundary value problem. Based on such symmetry, symmetric homotopies are constructed to efficiently compute all solutions of the resulting polynomial systems for problems with cubic nonlinearity and quintic nonlinearity, by applying the afore proved Chen-Xie conjecture and its analogue for—△u=u5, respectively. Since only representative solution paths are need to be followed, a lot of computational cost can be saved.5. Homotopy continuation method and damped Newton method are two known meth-ods for circumventing the drawback of local convergence of standard Newton method. Al-though some relations between these two methods have been obtained early, these relations are mainly on the differential equations which determine the paths followed by the two methods, rather than the sequences generated by the two algorithms. These sequences are investigated and some further relations are explored in terms of the marching directions and the stepsizes during the iteration processes.