For more information about this meeting, contact Fei Wang, Hope Shaffer, Jinchao Xu, Toan Nguyen.
|Title:||A finite element method for the second order linear elliptic equation in non-divergence form|
|Seminar:||CCMA PDEs and Numerical Methods Seminar Series|
|Speaker:||Wujun Zhang, University of Maryland|
|Fully nonlinear elliptic PDEs, including Monge-Ampere equation and Isaac's equation, arise naturally from differential geometry, optimal mass transport, stochastic games and the other fields of science and engineering. In contrast to an extensive PDE literature, the numerical approximation reduces to a few papers. One major difficulty is the notion of viscosity solution which hinges on the maximum principle, instead of a variational principle.
In this talk, we consider linear uniformly elliptic equations in non-divergence form, which may be regarded as linearization of fully nonlinear equations. We discuss the design of convergent numerical methods for such equations. We introduce a novel finite element method which satisfies a discrete maximum principle. This property, together with operator consistency, guarantees convergence to the viscosity solution provided that the coefficient matrix and right-hand side are continuous. We also present a discrete version of the Alexandroff-Bakelman-Pucci estimate, and use it to derive convergence rates under suitable regularity assumptions of the coefficient matrix and viscosity solution.|
Room Reservation Information
|Date:||04 / 18 / 2014|
|Time:||03:30pm - 05:00pm|