Partial differential equations, initial value and time-dependent initial- boundary value problems

Athena Picarelli
Associate Professor
Research interests
Topic People Description
MSC 65M06 - Finite di erence methods Athena Picarelli
Only in very few cases Hamilton-Jacobi-Bellman equations admit an explicit solution. It becomes then fundamental the numerical approximation of the solution. Numerical methods for partial differential equations are basically divided in: finite elements methods and finite difference methods. The latter are based on a Taylor approximation of derivatives. They are quite simple and intuitive methods for which a complete convergence analysis in the class of solutions of the equation in the viscosity sense is available.
MSC 65M15 - Error bounds Athena Picarelli
Defined a numerical approximation scheme for a partial differential equation and proved its convergence, it is also interesting to provide error estimates. For classical solutions of elliptic and parabolic equation this can be obtained by quite standard techniques. However, in the particular case of viscosity solutions specific analytic regularization techniques have to be applied.