2018/03/02 by Lloyd N. Trefethen, Trefethen, Lloyd N. · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Applied mathematics #Boundary (topology) #Boundary value problem #Computer science #Conformal map #Convergent series #Degree (music) #Exponential function #FOS: Mathematics #Laplace transform #Laplace's equation #Mathematical analysis #Mathematics #Numerical Analysis (math.NA) #Physics #Power series #Series (stratigraphy) #Set (abstract data type)
paper · pdf · doi:10.48550/arxiv.1803.00973
openalex publication_date 2018/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
At the ANZIAM conference in Hobart in February, 2018, there were several talks on the solution of Laplace problems in multiply connected domains by means of conformal mapping. It appears to be not widely known that such problems can also be solved by the elementary method of series expansions with coefficients determined by least-squares fitting on the boundary. (These are not convergent series; the coefficients depend on the degree of the approximation.) Here we give a tutorial introduction to this method, which converges at an exponential rate if the boundary data are sufficiently well-behaved. The mathematical foundations go back to Runge in 1885 and Walsh in 1929. One of our examples involves an approximate Cantor set with up to 2048 components.