Introduction

Any polygon, no matter how irregular, can be represented exactly as a union of triangles, and any polyhedron can be represented as a union of tetradehra. It is thus reasonable to employ the triangle as the fundamental element shape in two dimensions, and to extend a similar treatment to three-dimensional problems by using tetrahedra.

The solution accuracy obtained with simple elements may be satisfactory in some problems, but it can be raised markedly by using piecewise polynomials instead of piecewise-linear functions on each element. If desired, this increased accuracy can be traded for computing cost, by using high-order approximations on each element but choosing much larger elements than in the first-order method. Indeed, both theory and experience indicate that for many two-dimensional problems, it is best to subdivide the problem region into the smallest possible number of large triangles, and to achieve the desired solution accuracy by the use of highorder polynomial approximants on this very coarse mesh.

In the following, details will be given for the construction of simplicial elements – triangles and tetrahedra – for the inhomogeneous scalar Helmholtz equation. This equation is particularly valuable because of its generality; a formulation valid for the inhomogeneous Helmholtz equation allows problems in Laplace's equation, Poisson's equation, or the homogeneous Helmholtz equation to be solved by merely dropping terms from this general equation. Scalar and quasi-scalar problems will be considered throughout, while all materials will be assumed locally linear and isotropic.