A Simple Test Problem for Multivariate Quadrature

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 x=(x_1,x_2)\in [-1,1]^2. Given parameters \delta_1>1 and \delta_2>1, define the function

f(x) = \frac{1}{(x_1-\delta_1)(x_2-\delta_2)}.

Notice that f is monotonic, and f(x)>0. The parameters \delta_1 and \delta_2 determine how quickly f grows near the boundary. The closer \delta_1 and \delta_2 are to 1, the closer the singularity in the function gets to the domain, which determines how large f is at the point (x_1=1,x_2=1). Anisotropy can be introduced in the function by choosing \delta_1\not=\delta_2. Based on experience, I’ve seen that 1<\delta_1,\delta_2<1.1 are difficult functions to approximate with polynomials, while \delta_1,\delta_2\approx 2 will be a very easy functions to approximate with polynomials. For any choice of \delta_1,\delta_2>1, the function f is analytic in x, 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 \delta_1 and \delta_2.

I’ll update this with some Matlab examples shortly.


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