MA587: Numerical PDE: Finite Element Method, Spring 2022

Instructor:  Dr. Zhilin Li                                         Time: T TH : 4:30pm-5:45pm
Office:  SAS 3148, Tel: 515-3210
E-mail:  <click to e-mail >

The Finite Element Method is one of very important numerical methods for solving partial differential equations in science and engineering. In this introductory course, we will start with theoretical foundations and algorithm implementations for one-dimensional problems so that we can learn the essential tools that carry over to higher dimensions. The discussion of two-dimensional problems, some common used finite element spaces, error analysis, and other related topics including some applications of the finite element method will then be followed. Efforts will also be made on the issues of implementation and related software packages. Using the data from the Matlab mesh generation, the students will be able to implement finite element method using  their favorite computer languages.

Grading:

Homework/project  (analytic part and computer projects, may include in-class work) about every two weeks:  One assignment can be replaced by a project of students' own research work. Special arrangements can be arranged if necessary.
 
Extra credit: can be earned for actively involved in class with questions/interactions, or extra credit problems in the assignments.

• Numerical Solutions of Partial Differential Equations– An Introduction to Finite Difference and Finite Element Methods by Cambridge (available online at NCSU library)
  
• C. Johnson, Numerical solution of partial differential equations by the finite element method

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Materials:

• Introduction
• FEM for 1D elliptic problems
  
• Weak form, minimization problems, and equivalencies
• Basis functions
• Assembly
• Theoretical basis, Sobolev space, Error analysis
• Sturm-Liouvillle problem
• High order elements
• High order equation
• Finite element code and data structure
• FEM for 2D elliptic problems
• Weak form and boundary conditions
• Some finite element space
• Interpolation and error estimates
• FEM programming
• FEM for parabolic equation and hyperbolic equations
• Matlab tool-box and lab
• Discontinuous Galerkin method
• Applications of FEM

A few selected references:

• Understanding and Implementing the Finite Element Method by Mark S. Gockenbach, SIAM, 2006.
• Finite Element, I-V, G. F. Carey and J. T. Oden, Prentice-Hall, Inc, Englewood Cliffs, 1983.
• Partial Differential Equation:  (Matlab) Toolbox, The Math-Works Inc.
  
• An Analysis of the Finite Element Method, Second Edition by Gilbert Strang and George Fix, Wellesley-Cambridge Press 

Disability services: