Error estimators and adaptivity for the neutron diffusion problem

From an Object-Oriented analysis, a new nodal method was developed for the solution of the neutron diffusion equation, consisting of the minimization and equidistribution of a total error estimator ET by the evolution of a dynamical gradient system of the form dΦ/dτ = −ΔET(Φ(τ)) on the manifold given by double vertical barΦdouble vertical bar = constant. The analysis discriminates a class of functions and a class of operators; in the class of functions, bases of polynomials of increasing number of terms were defined, and some procedures related to the class were implemented, such as rotations, differentiations, and two inner products. The new method was applied to the solution of the static one-speed diffusion equation on regular polygons and on a 2-D heterogeneous hexagonal domain. The results showed that the minimization of errors by the gradient system exhibits a monotone iterative behavior, controlling the distance between the approximate and the exact solution, not merely the distance between the previous and present step solutions. Adaptivity criteria appear naturally from the error observation, suggesting how and where it is necessary to improve the solution. Finally, it is discussed how the method (and the Object-Orientation frame) allows the relievement of users from uncertainties and the conservation of the accuracy without the user intervention.