This course will be provide an overview of two standard numerical methods for partial differential equations (PDEs). The focus will be on essential theoretical analysis as well parabolic and hyperbolic partial differential equations. This will be followed by a short foray into linear system solvers and finite difference scheme for two-dimensional Poisson's (elliptic) problem. The second part of the course will deal with finite element methods exclusively for elliptic problems. The core ideas in functional analysis, variational formulation, error analysis, and computer implementation will be presented for the one-dimensional problem. This will be followed be a more practical treatment of two-dimensional problems. The last part will consist of an overview of the specialized topics of mixed and adaptive finite element methods.as computer implementation. The first part will be on finite difference methods. Key numerical schemes and underlying theory will be provided for one-dimensional.
Students learn about basics of partial differential equations, some qualitative and quantitative aspects of their analytical and general solutions.
Students can derive and analyze finite difference methods for solving model problems of parabolic and hyperbolic partial differential equations.
Students can write computer code to solve model problems, interpret and visualize their solutions. They also learn to make appropriate choices for numerical linear algebraic methods.
Students understand basics of functional analysis, variational formulation and finite element spaces. They also learn stability analysis in the setting of Hilbert spaces.