Course Category：Professional Basic Course
Prerequisites：Fundamentals of PDEs, computational methods of differential equations, numerical algebra (or Matrix theory), functional analysis.
Aims & Requirements：
This course is a speciality foundation course for Master degree candidates majoring in computational mathematics and applied mathematics. This course, in terms of finite element method, offers comparatively comprehensive mathematical foundation, and introduces some up-to-date researches concerning finite element. Detailed explanations will be offered to all mathematical contents which are not included in undergraduate curriculum. For instance, comprehensive results will be elaborated for Sobolev space theory and nonlinear functional analysis but detailed proofs will not be given, except for elementary proofs for the simpler special cases, so that learners of the course can have a better understanding. Participants of the course are expected to have a command of the basic theories and methods of finite element method and lay a solid foundation for future work and research.
Chapter 1 Variational Principle and Sobolev Space
Minimization of differentiable quadratic convex functionals; minimization of non-differentiable convex functionals; Sobolev spaces; imbedding theorem; trace theorems.
Chapter 2 Elliptic boundary value problems
Second order elliptic boundary value problem; boundary value problem of linear elasticity; fourth order elliptic boundary value problem.
Chapter 3 Finite element discretization
Basis properties; triangular elements; rectangular elements; conforming finite elements of fourth order problems.
Chapter 4 Conforming finite element methods
Generalities of convergence; piece-wise polynomial interpolation in Sobolev spaces; finite element error analysis of second order problems on polygonal regions; inverse inequalities in finite element spaces.
Chapter 5 Non-conforming finite element methods
Abstract error estimate; non-conforming elements for second order problems; non-conforming elements for fourth order problems.
Chapter 6 Mixed finite element methods
Mixed variational formulation; abstract theory; mixed finite element methods for second order problems; mixed finite element method for Stokes problem.
Chapter 7 Adaptive finite element methods
Backward error estimator of residual type; reliability and efficiency; convergence of algorithms.
Chapter 8 Multi-level methods and domain decomposition methods
Basic ideas of multi-mesh methods; convergence of multi-mesh methods; layer based methods; introduction to domain decomposition methods; overlapping domain decomposition methods; DN alternating methods.
WANG Lieheng, XU Xuejun, Mathematical Foundations of Finite Element Methods, Science Press, Beijing, 2004.
 P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978.
 S.C. Brenner and C.R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, 1994.
 D. Braess, Finite Elements, Cambridge University Press, 2001.
 J.H. Bramble, Multigrid Methods, Pitman, 1993.
Author：Lieheng Wang, Xuejun Xu (Academy of Mathematics and Systems Science)