The orthogonal decomposition theorems for mimetic finite difference methods

Accurate discrete analogs of differential operators that satisfy the identi ties and theorems of vector and tensor calculus provide reliable finite dif ference methods for approximating the solutions to a wide class of partial differential equations. These methods mimic many fundamental properties of the underlying physical problem including conservation laws, symmetries in the solution, and the nondivergence of particular vector fields (i.e., they are divergence free) and should satisfy a discrete version of the orthogon al decomposition theorem. This theorem plays a fundamental role in the theo ry of generalized solutions and in the numerical solution of physical model s, including the Navier-Stokes equations and in electrodynamics. We are der iving mimetic finite difference approximations of the divergence, gradient, and curl that satisfy discrete analogs of the integral identities satisfie d by the differential operators. We first define the natural discrete diver gence, gradient, and curl operators based on coordinate invariant definitio ns, such as Gauss's theorem, for the divergence. Next we use the formal adj oints of these natural operators to derive compatible divergence, gradient, and curl operators with complementary domains and ranges of values. In thi s paper we prove that these operators satisfy discrete analogs of the ortho gonal decomposition theorem and demonstrate how a discrete vector can be de composed into two orthogonal vectors in a unique way, satisfying a discrete analog of the formula (A) over right arrow grad phi + curl (B) over right arrow. We also present a numerical example to illustrate the numerical proc edure and calculate the convergence rate of the method for a spiral vector field.