Solving the Laplace equation on the disc using the UAT spline

In this work, we are interested in the resolution of the Laplace equation -Delta u=f with Dirichlet boundary condition in a closed surface S in R-2, which is - topologically - equivalent to the unit disc D={(x, y) |x(2)+y(2) <= 1}. It is known that for a function u represented in polar coordinates on D, certain boundary conditions must be satisfied by u so that the surface S is of class C-0. More precisely, we construct an approximant of class C-0 on D as a tensor product of two quasi-interpolants, one based on UAT-splines and the other based on classical B-splines. Some numerical results are given to validate the work.