Have you ever solved a problem numerically and wondered: How accurate is my answer? Oftentimes the only clue you have is how the solution changes (hopefully not much) as you refine the mesh and solve it with more degrees of freedom. I have posted some new problems and a few solutions that help with this problem.
The method is to use mathematical theorems that bound the error. Sometimes the error can only be bounded in an average sense – called the mean square error, which is the error squared, integrated over the domain, and then take the square root. The best situation, though, is when you can bound the difference between the exact solution and the approximate solution you’ve found at every point in the domain. That is called the point-wise error.
The book Method of Weighted Residuals and Variational Principles discusses these points in Chapter 11. While the book was first published in 1972, it has been republished by the Society for Industrial and Applied Mathematics, http://www.ChemEComp.com/MWR. Of course the mathematical proofs are still valid. But, back in 1972, you couldn’t solve many problems very accurately. However, now you can. Thus, the problems I’ve posted on the website use the theorems given or mentioned in the book and solve the problems using modern technology, and then compute the mean-square error bound and sometimes the point-wise error bound for the approximate solution. The solutions are sometimes found using MATLAB and sometimes found with Comsol Multiphysics.
Problems 7, 8, and 9 have just been posted. These refer to a problem called ‘effectiveness factor’ in chemical engineering. They compute the average reaction rate in a porous catalyst that is undergoing both diffusion and reaction. This is a classic problem in chemical reaction engineering. Problem 7 sets the stage – gives the theorems and equations that aren’t already in the book and solves the problem for a plane geometry using Hermite cubic polynomials in Comsol Multiphysics. Other parts use Lagrange quadratic polynomials and solve for both plane geometry and a spherical catalyst pellet. Problem 9 is for a non-linear problem. (You thought no error bounds would be available didn’t you!)