Applications of the Variational Integral in Iterative Numerical Solutions to the Stationary Heat Equations

摘要

Three stationary heat equations were solved using the finite-difference method. The resulting set of algebraic equations were solved using the Gauss-Seidel iterative technique. The temperatures at the nodal points were substituted into a numerical approximation of the variational integral. The variational integral approximation was used to determine when to stop the iterative process. The variational integral stopping criterion was compared to a stopping criterion that uses the displacement between iterations to approximate the error between the iterative solution and the exact solution. The variational integral was found to be less effective as a stopping criterion than the error estimate. The variational integral was examined as a method of determining whether the finite-difference technique or the finite-element technique gave a more accurate solution. It was found that the variational integral failed, in some cases, to predict the more accurate method. (Author)