I’ve found it difficult to find really simple test problems for multivariate quadrature ideas. Here’s one that I use over and over. It’s useful for testing Gaussian type quadrature rules and polynomial approximation. Its solution contains many of the features seen in the solution of an elliptic PDE with parameterized coefficients and a constant forcing term.
Let . Given parameters and , define the function
Notice that is monotonic, and . The parameters and determine how quickly grows near the boundary. The closer and are to 1, the closer the singularity in the function gets to the domain, which determines how large is at the point . Anisotropy can be introduced in the function by choosing . Based on experience, I’ve seen that are difficult functions to approximate with polynomials, while will be a very easy functions to approximate with polynomials. For any choice of , the function is analytic in , so we expect polynomial approximations to converge exponentially as the degree of approximation increases. However, the rate of exponential convergence and the number of polynomial terms required to achieve a given error tolerance will depend heavily on and .
I’ll update this with some Matlab examples shortly.