By Mark S. Gockenbach (siam, 2010)

Creating a mesh

The data structure is described in the M-file Mesh1.m (this M-file contains only comments, which are displayed when you type "help Mesh1"):

help Mesh1

The Fem1 package implements piecewise linear finite elements for two

dimensional problems and is intended to accompany "Partial

Differential Equations: Analytical and Numerical Methods" (second

edition) by Mark S. Gockenbach (SIAM 2010). It uses the data

structure described in Section 13.1 of the text (that is, the four

arrays NodeList, NodePtrs, FNodePtrs, and ElList, augmented by

three more arrays (CNodePtrs, ElEdgeList, and FBndyList) to allow

the code to handle inhomogeneous boundary conditions (as hinted at

in the text). These seven arrays are collected into a structure.

NodeList: Mx2 matrix, where M is the number of nodes in the mesh

(including boundary nodes). Each row corresponds to a

node and contains the coordinates of the node.

NodePtrs: Mx1 matrix. Each entry corresponds to a node:

if node i is free, NodePtrs(i) is the index of node i

in FNodePtrs; else NodePtrs(i) is the negative of

index of node i in CNodePtrs

FNodePtrs: Nx1 vector, where N is the number of free nodes.

NodePtrs(i) is the index into NodeList of the ith

free node. So, for example, NodeList(NodePtrs(i),1:2)

are the coordinates of the ith free node.

CNodePtrs: Kx1 vector, where K=M-N is the number of constrained

nodes. CNodePtrs(i) is the index into NodeList of the

ith constrained node.

ElList: Lx3 matrix, where L is the number of triangular elements

in the mesh. Each row corresponds to one element and

contains pointers to the nodes of the triangle in

NodeList.

ElEdgeList: Lx3 matrix. Each row contains flags indicating

whether each edge of the corresponding element is on

the boundary (flag is -1 if the edge is a constrained

boundary edge, otherwise it equals the index of the

edge in FBndyList) or not (flag is 0). The edge

of the triangle are, in order, those joining vertices

1 and 2, 2 and 3, and 3 and 1.

FBndyList: Bx2 matrix, where B is the number of boundary edges

not constrained by Dirichlet conditions. Each

row corresponds to one edge and contains pointers

into NodeList, yielding the two vertices of the edge.

assuming Dirichlet conditions on the boundary.

T=RectangleMeshD1(nx,ny,lx,ly)

This function creates a regular, nx by ny finite element mesh for

a Dirichlet problem on the rectangle [0,lx]x[0,ly].

The last three arguments can be omitted; their default values are

ny=nx, lx=1, ly=1. Thus, the command "T=RectangleMeshD1(m)"

creates a regular mesh with 2m^2 triangles on the unit square.

For a description of the data structure describing T, see

"help Mesh1".
The command RectangleMeshN1 creates a regular triangulation of the same rectangle, assuming Neumann conditions on the boundary, while RectangleMeshTopD1 imposes Dirichlet conditions on the top edge and Neumann conditions on the other three edges.
Thus I only provide the means to deal with a single domain shape (a rectangle), and only under three combinations of boundary conditions. To use my code to solve BVPs on another domain, you will have to write code to generate the mesh yourself. Note, however, that the rest of the code is general: it handles any polygonal domain, with any combination of homogeneous Dirichlet and Neumann conditions.
Here I create a mesh:
clear

T=RectangleMeshD1(4)

T =

NodeList: [25x2 double]

FNodePtrs: [9x1 double]

CNodePtrs: [16x1 double]

ElList: [32x3 double]

ElEdgeList: [32x3 double]

FBndyList: []

ShowMesh1(T,flag)

This function displays a triangular mesh T. Unless the flag

is nonzero, free nodes are indicated by a 'o', and

constrained nodes by an 'x'.

For a description of the data structure T, see "help Mesh1".

The optional argument flag has the following effect:

flag=1: the triangles are labeled by their indices

flag=2: the nodes by their indices

flag=3: both the nodes and triangles are labeled

flag=4: the free nodes are labeled by their indices.

flag=5: the free and constrained nodes are labeled by