On the Construction and Convergence of a Finite-Element Solution of Laplace's Equation

We propose an algorithm for constructing automatically finite-element approximations to Laplace's equation. Thisuses triangular elements which have internal angles that are never very small and have sides that cannot be pathologically small or pathologically large; it ensures that the matrix of the system is monotone. We assume that the boundary is twice-continuously differentiable but may consist of more than one closed arc, so that the region it encloses is not necessarily simply-connected. Assuming that the solution is twice-continuously differentiable on the region and its boundary, we prove 0(h) convergence in the Sobolev $W2(1)$ norm.