We present an algorithm for finding high order numerical approximations of minimal surfaces with a fixed boundary. The algorithm employs parametrization by high order polynomials and a linearization of the weak formulation of the Laplace-Beltrami operator to arrive at an iterative procedure to evolve from a given initial surface to the final minimal surface. For the steady state solution we measure the approximation error in a case where the exact solution is known (the catenoid). In the framework of parametric interpolation, the choice of interpolation points (mesh nodes) is directly affecting the approximation error, and we discuss how to best update the mesh on the evolutionary surface such that the parametrization remains smooth. In our test case we achieve exponential convergence in the approximation of the minimal surface as the polynomial degree increases, but the rate of convergence greatly differs with different choices of mesh update algorithms.

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In this paper we discuss spectral approximations of the Poisson equation in deformed quadrilateral domains. High order polynomial approximations are used for both the solution and the representation of the geometry. Following an isoparametric approach, the four edges of the computational domain are first parametrized using high order polynomial interpolation. Transfinite interpolation is then used to construct the mapping from the square reference domain to the physical domain. Through a series of numerical examples we show the importance of representing the boundary of the domain in a careful way; the choice of interpolation points along the edges of the physical domain may significantly effect the overall discretization error. One way to ensure good interpolation points along an edge is based on the following criteria: (i) the points should be on the exact curve; (ii) the derivative of the exact curve and the interpolant should coincide at the internal points along the edge. Following this approach, we demonstrate that the discretization error for the Poisson problem may decay exponentially fast even when the boundary has low regularity.

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