APCOM & ISCM 11-14 th December, 2013, Singapore 1 Graph grammar based multi-frontal direct solver for isogeometric FEM simulations on GPU *M. Paszyński¹, K. Kuźnik 1 , V.M. Calo 2 and D. Pardo 3 1 AGH University of Science and Technology, Krakow, Poland. 2 King Abdullah University of Science and Technology, Thuwal, Saudi Arabia. 3 The University of the Basque Country, UPV/EHU and Ikerbasque, Bilbao, Spain. *Corresponding author: maciej.paszynski@AGH.EDU.PL Abstract We present a multi-frontal direct solver for two dimensional isogeometric finite element method simulations with NVIDIA CUDA and perform numerical experiments for linear, quadratic and cubic B-splines. We compare the computational cost O(Np 2 ) for 2D parallel shared memory implementation with the corresponding estimate O(N 1.5 p 3 ) for a standard 2D sequential implementation. We conclude the presentation with observation that computational cost of the shared memory direct solver scales like p 2 when we increase the global continuity of the isogeometric solution, which is an adventage with respect to sequential isogeometric solver scalability of the order of p 3 . Keywords: Multi-frontal direct solver, isogeometric finite element method, computational cost, shared memory machine Introduction The isogeometric finite element method (Cottrel et al. 2009) is a higher order method providing global C k continuity of the solution. It is based on the usage of B-spline basis functions delivering higher order global regularity of the solution. The classical higher order finite element method (Demkowicz 2006, Demkowicz et al. 2007) provides C 0 global continuity only. The isogeometric finite element method generates a sparse system of equations that can be solved by multi-frontal direct solver algorithm (Duff et al. 1984, Duff et al. 1983, Geng et al. 2006). In this paper we present how isogeometric C k finite element method multi-frontal solver differs from C 0 higher order finite element method solver by factor of p 3 .We also show how this p 3 factor can be reduce down to p 2 factor by using shared memory implementation. B-spline based isogeometric finite element method We focus on the 2D model problem, namely the Laplace equation over a square domain   0 , 2 1   x x u for     2 2 1 1 , 0 ,    x x (1)   0 , tr 2 1  x x u for   , 1 , 0 1  x 0 2  x (2)   1 , tr 2 1  x x u for   , 1 , 0 1  x 1 2  x (3)   0 , 2 1 1    x x x u for   , 1 , 0 1  x   1 , 0 2  x (4) where         1 , 0 , 1 , 0 : , 2 1 2 1     x x x x D (5)         1 , 0 , 1 , 0 : , 2 1 2 1     x x x x N (6)