MAXIMUM PRINCIPLES FOR P1-CONFORMING FINITE ELEMENT APPROXIMATIONS OF QUASI-LINEAR SECOND ORDER ELLIPTIC EQUATIONS

This paper derives some discrete maximum principles for P1-conforming finite element approximations for quasi-linear second order elliptic equations. The results are extensions of the classical maximum principles in the theory of partial differential equations to finite element methods. The mathematical tools are based on the variational approach that was commonly used in the classical partial differential equation theory. The discrete maximum principles are established by assuming a property on the discrete variational form that is of global nature. In particular, the assumption on the variational form is verified when the finite element partition satisfies some angle conditions. For the general quasi-linear elliptic equation, these angle conditions indicate that each triangle or tetrahedron needs to be O(h(alpha))-acute in the sense that each angle alpha(ij) (for the triangle) or interior dihedral angle a(ij) (for the tetrahedron) must satisfy alpha(ij) <= pi/2 - gamma h(alpha) for some alpha >= 0 and gamma > 0. For the Poisson problem where the differential operator is given by the Laplacian, the angle requirement is the same as the existing ones: either all the triangles are nonobtuse or each interior edge is nonnegative. It should be pointed out that the analytical tools used in this paper are based on the powerful De Giorgi iterative method that has played important roles in the theory of partial differential equations. The mathematical analysis itself is of independent interest in the finite element analysis.