Regularization Methods for Differential Equations and Their Numerical Solution

The objectives of the thesis are to propose and to investigate approximate methods for various differential equations with or without constraints. Most attention is paid to ordinary and partial differential equations with constraints (where the solution is know to lie in an explicitly defined invariant manifold). We propose and analyze a regularization method called sequential regularization method (SRM) and its numerical approximation. A very important improvement of the SRM over usual regularization methods is that the problem after regularization need not be stiff. Hence explicit difference schemes can be used to avoid solving nonlinear systems. This makes the computation much simpler. The method is applied in several application fields such as mechanical constrained multi-body systems, nonstationary incompressible Navier-Stokes equations which is an example of partial differential equations with constraints (PDAE), and miscible displacement in porous media in reservoir simulation. Improvements over stabilization methods that stabilize the invariant manifold over long time intervals and extra benefits for these applied problems are also achieved. We finally discuss the numerical solution of several singular perturbation problems which come from many applied areas and regularized problems. Uniformly convergent schemes with respect to the perturbation parameter are constructed and proved. A spurious solution phenomenon for an upwinding scheme is analyzed.