Emine Celiker

Bio: Emine Celiker is an academic researcher from University of Dundee. The author has contributed to research in topics: Finite element method & Nonlinear system. The author has an hindex of 1, co-authored 1 publications receiving 3 citations.

A new numerical approach to the solution of the 2-D Helmholtz equation with optimal accuracy on irregular domains and Cartesian meshes

Abstract: A new numerical approach for the time independent Helmholtz equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. At similar 9-point stencils, the accuracy of the new approach is two orders higher for the Dirichlet boundary conditions and one order higher for the Neumann boundary conditions than that for the linear finite elements. The numerical results for irregular domains also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. The new approach can be equally applied to the Helmholtz and screened Poisson equations.

The numerical solution of the 3D Helmholtz equation with optimal accuracy on irregular domains and unfitted Cartesian meshes

Abstract: Here, we extend the optimal local truncation error method (OLTEM) recently developed in our papers to the 3D time-independent Helmholtz equation on irregular domains. Trivial unfitted Cartesian meshes and simple 27-point discrete stencil equations are used for 3D irregular domains. The stencil coefficients for the new approach are assumed to be unknown and are calculated by the minimization of the local truncation error of the stencil equations. This provides the optimal order of accuracy of the proposed technique. At similar 27-point stencils, the accuracy of OLTEM is two orders higher for the Dirichlet boundary conditions and one order higher for the Neumann boundary conditions compared to that for linear finite elements. The numerical results for irregular domains also show that at the same number of degrees of freedom, OLTEM is even much more accurate than high-order (up to the fifth order) finite elements with much wider stencils. Compared to linear finite elements with similar 27-point stencils, at accuracy of 0.1% OLTEM decreases the number of degrees of freedom by a factor of greater than 1000. This leads to a huge reduction in computation time. The new approach can be equally applied to the Helmholtz and screened Poisson equations.

On the numerical solutions of two-dimensional scattering problems for an open arc

Abstract: For the scattering problems of acoustic wave for an open arc in two dimensions, we give a uniqueness and existence analysis via the single layer potential approach leading to a system of integral equations that contains a weakly singular operator. For its numerical solutions, we describe an $O(h^{3})$ order quadrature method based on the specific integral formula including convergence and stability analysis. Moreover, the asymptotic expansion of errors with odd power $O(h^{3})$ is got and the Richardson extrapolation algorithm (EA) is used to improve the accuracy of numerical solutions. The efficiency of the method is illustrated by a numerical example.