Tetrahedral polynomial finite elements for the Helmholtz equation

It is shown that the Helmholtz equation in three dimension leads to finite element approximations on tetrahedral elements that closely resemble the corresponding two-dimensional treatment on triangle. For each polynomial order, there exist two numeric universal matrices independent to tetrahedron size and shape; the element matrices are always given as linear combinations of row and column permutations of these. Numeric matrices are given up to third-order, and the permutation schemes are shown in detail. Experimental computer programs using these elements have shown fast matrix assembly times; convergence rates are essentially similar to those obtained with the corresponding triangular elements.