HIGH-ORDER POLYNOMIAL TRIANGULAR FINITE ELEMENTS FOR POTENTIAL PROBLEMS

An analytic derivation is given for high-accuracy triangular finite elements useful for numerical solution of field problems involving Laplace's, Poisson's, Helmholtz's, or related elliptic partial differential equations in two dimensions. General expressions are developed for complete polynomial fields of arbitrarily high order, and the method for obtaining element describing matrices is shown. These matrices can always be written in terms of trigonometric functions of the vertex angles, and the triangle area, multiplied by certain numerical coefficient matrices which are the same for any triangle. For polynomial fields up to fourth order, the numerical coefficient matrices are given, so that the element matrices for any triangle can be found easily. Use of these new elements is illustrated by a simple vibration problem. © 1969.