Three families of $$C^1$$ - $$P_{2m+1}$$ Bell finite elements on triangular meshes

The \(C^1\)-\(P_5\) Bell finite element removes the three degrees of freedom of the edge normal derivatives of the \(C^1\)-\(P_5\) Argyris finite element. We call a \(C^1\)-\(P_k\) finite element a Bell finite element if it has no edge-degree of freedom and it contains the \(P_{k-1}\) space locally. We construct three families of odd-degree \(C^1\)-\(P_{2m+1}\) Bell finite elements on triangular meshes. Comparing to the \(C^1\)-\(P_{2m}\) Argyris finite element, the \(C^1\)-\(P_{2m+1}\) Bell finite elements produce same-order solutions with much less unknowns. For example, the second \(C^1\)-\(P_7\) Bell element (from the second family) and the \(C^1\)-\(P_6\) Argyris element have numbers of local degrees of freedom of 31 and 28 respectively, but oppositely their numbers of global degrees of freedom are 12V and 19V asymptotically, respectively, where V is the number of vertices in a triangular mesh. A numerical example says the new element has about 3/4 number of unknowns, but is about 5 times more accurate. Numerical computations with all three families of elements are performed.