Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I:: the scalar case

We develop a one-parameter family of hp-version discontinuous Galerkin finite element methods, parameterised by theta is an element of [-1, 1], for the numerical solution of quasilinear elliptic equations in divergence form on a bounded open set Omega subset of R-d, d >= 2. In particular, we consider the analysis of the family for the equation -del .{mu(x, vertical bar del u vertical bar)del u} = f(x) subject to mixed Dirichlet-Neumann boundary conditions on partial derivative Omega. It is assumed that mu is a real-valued function, mu is an element of C((Omega) over bar x [0, infinity)), and there exist positive constants m(mu) and M-mu such that m(mu)(t - s) <= mu(x, t)t - mu(x, s)s <= M-mu(t - s) for t >= s >= 0 and all x is an element of (Omega) over bar. Using a result from the theory of monotone operators for any value of theta is an element of [-1, 1], the corresponding method is shown to have a unique solution u(DG) in the finite element space. If u is an element of C-1(Omega) boolean AND H-k(Omega), k >= 2, then with discontinuous piecewise polynomials of degree p >= 1, the error between u and u(DG), measured in the broken H-1(Omega)-norm, is O(h(s-1)/p(k-3/2)), where 1 <= s <= min {p + 1, k}.