A Fem for the Square Root of the Laplace Operator: Master’s Scholarly Paper Applied Mathematics and Scientific Computation Program, Department of Mathematics, University of Maryland

Abstract. We consider the square root of the Laplace operator (−∆)1/2 in a bounded domain. The square root of the Laplacian can be realized as the Dirichlet to Neumann operator of an extension problem posed on a semi-infinite cylinder. This extension problem involves a mixed boundary value problem, which we analyze in the framework of Sobolev spaces. For numerical approximation we propose a suitable truncated problem, which can be justified by the rapid decay of the solution to the extension problem. A finite element approximation is considered for the truncation. A priori error estimates are obtained for conforming and shape regular meshes and numerical experiments are presented illustrating the theory.