Elliptic Equations: Equilibrium in Two Dimensions
This chapter formulates the finite-element method to solve a function that depends on two space dimensions. It starts with very coarse meshes so that assembly of the element matrices, applications of boundary conditions, and solving for the unknown primary and secondary variables become amenable to hand calculations or with the help of a calculator. The chapter develops MATLAB codes for mesh refinement and error assessment. It discusses the isoparametric elements and solve problems over domains with curved boundaries and complex problem parameters. The chapter also develops MATLAB codes for such problems including mesh generation, numerical evaluation of element matrices, assembly of elements, applications of boundary conditions, solving for primary variables and post-processing to obtain quantities of interest. It considers problems with simple boundaries such that the domain can be easily divided into triangles and/or rectangles whose sides are parallel to the coordinate axes.