Construction and geometric validity (positive jacobian) of serendipity Lagrange finite elements, theory and practical guidance

Finite elements of degree two or more are needed to solve various P.D.E. problems. This paper discusses a method to validate such meshes for the case of the serendipity Lagrange elements of various degree. The first section of this paper comes back to Bézier curve and Bézier patches of arbitrary degree. The way in which a Bézier patch and a complete finite element are related is recalled. The construction of serendipity or reduced Lagrange finite elements of various degree is discussed, including simplices (triangle and tetrahedron), quads, prisms (pentahedron), pyramids and hexes. The validity condition, the positivity of the jacobian, exhibited for the classical (complete) elements is used to validate their serendipity counterparts after having invented a complete element equivalent to the reduced element under analyse.