Browsing by Author "Gürkan, Ceren"

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Citation Count: 0
eXtended Hybridizable Discontinuous Galerkin (X-HDG) method for linear convection-diffusion equations on unfitted domains
(Academic Press inc Elsevier Science, 2024) Gürkan, Ceren; Gurkan, Ceren
In this work, we propose a novel strategy for the numerical solution of linear convection diffusion equation (CDE) over unfitted domains. In the proposed numerical scheme, strategies from high order Hybridized Discontinuous Galerkin method and eXtended Finite Element method are combined with the level set definition of the boundaries. The proposed scheme and hence, is named as eXtended Hybridizable Discontinuous Galerkin (XHDG) method. In this regard, the Hybridizable Discontinuous Galerkin (HDG) method is eXtended to the unfitted domains; i.e., the computational mesh does not need to fit to the domain boundary; instead, the boundary is defined by a level set function and cuts through the background mesh arbitrarily. The original unknown structure of HDG and its hybrid nature ensuring the local conservation of fluxes is kept, while developing a modified bilinear form for the elements cut by the boundary. At every cut element, an auxiliary nodal trace variable on the boundary is introduced, which is eliminated afterwards while imposing the boundary conditions. Both stationary and time dependent CDEs are studied over a range of flow regimes from diffusion to convection dominated; using high order (p <= 4) XHDG through benchmark numerical examples over arbitrary unfitted domains. Results proved that XHDG inherits optimal (p + 1) and super (p + 2) convergence properties of HDG while removing the fitting mesh restriction.
Citation Count: 1
Performance analyses of mesh-based local Finite Element Method and meshless global RBF Collocation Method for solving Poisson and Stokes equations
(Elsevier, 2022) Gürkan, Ceren; Gurkan, Ceren; Avci, Cem
Steady and unsteady Poisson and Stokes equations are solved using mesh dependent Finite Element Method and meshless Radial Basis Function Collocation Method to compare the performances of these two numerical techniques across several criteria. The accuracy of Radial Basis Function Collocation Method with multiquadrics is enhanced by implementing a shape parameter optimization algorithm. For the time-dependent problems, time discretization is done using Backward Euler Method. The performances are assessed over the accuracy, runtime, condition number, and ease of implementation. Three error kinds considered; least square error, root mean square error and maximum relative error. To calculate the least square error using meshless Radial Basis Function Collocation Method, a novel technique is implemented. Imaginary numerical solution surfaces are created, then the volume between those imaginary surfaces and the analytic solution surfaces is calculated, ensuring a fair error calculation. Lastly, all results are put together and trends are observed. The change in runtime vs. accuracy and number of nodes; and the change in accuracy vs. the number of nodes is analyzed. The study indicates the criteria under which Finite Element Method performs better and conditions when Radial Basis Function Collocation Method outperforms its mesh dependent counterpart.(c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
Citation Count: 12
Stabilized cut discontinuous galerkin methods for advection-reaction problems
(Society for Industrial and Applied Mathematics Publications, 2020) Gürkan, Ceren; Sticko, Simon; Massing, André
We develop novel stabilized cut discontinuous Galerkin methods for advection-reaction problems. The domain of interest is embedded into a structured, unfitted background mesh in \BbbR d where the domain boundary can cut through the mesh in an arbitrary fashion. To cope with robustness problems caused by small cut elements, we introduce ghost penalties in the vicinity of the embedded boundary to stabilize certain (semi-)norms associated with the advection and reaction operator. A few abstract assumptions on the ghost penalties are identified enabling us to derive geometrically robust and optimal a priori error and condition number estimates for the stationary advection-reaction problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. The theoretical results are corroborated by a number of computational studies for various approximation orders and for two- and three-dimensional test problems.