Solving the Laplace equation on the disc using the UAT spline

Abstract

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 $D=\left\{\left(x,y\right)\right|x^2+y^2⩽1\}$. 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.