Computational Physics

426 The finite element method for partial differential equations

P

Aa

Ab

Ac

1

2

3

Figure 13.2. The areasAa,AbandAcfor any pointPwithin the triangle. TheAi

are used to define the natural coordinatesξiof the pointP.Ais the total surface

area.

others. Any pointPwithin the triangle can be defined by specifying any two out of
three natural coordinates,ξa,ξborξc. These are defined by

ξi=

Ai

A

, i=a,b,c, (13.8)

whereAi,Aare the surface areas shown inFigure 13.2. The natural coordinates
satisfy the requirement
ξa+ξb+ξc=1. (13.9)

Thex- andy-coordinates of a point can be obtained from the natural coordinates
by the linear transformation



1

x

y



=





111

xa xb xc

ya yb yc









ξa

ξb

ξc



 (13.10)

where(xa,ya)are the Cartesian coordinates of vertexaetc.
The reverse transformation



ξa

ξb

ξc



=^1

2 A





xbyc−xcyb ybc xcb

xcyb−xbyc yca xac

xayb−xbya yab xba









1

x

y



, (13.11)

with 2A=det(A)=xbayca−xcaybaandxab=xb−xaetc., translates thex,y
coordinates into natural coordinates. All these relations can easily be checked.
Having natural coordinates, we can construct a piecewise linear approximation to
the solution from the values of the solution at the vertices of the triangles. Calling