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We propose the use of highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers. Starting from a highly continuous Isogeometric Analysis (IGA) discretization, we introduce C 0 -separators to reduce the interconnection between degrees of freedom in the mesh. By doing so, both the solution time and best approximation errors are simultaneously improved. We call the resulting method "refined Isogeometric Analysis (rIGA)". To illustrate the impact of the continuity reduction, we analyze the number of Floating Point Operations (FLOPs), computational times, and memory required to solve the linear system obtained by discretizing the Laplace problem with structured meshes and uniform polynomial orders. Theoretical estimates demonstrate that an optimal continuity reduction may decrease the total computational time by a factor between p 2 and p 3 , with p being the polynomial order of the discretization. Numerical results indicate that our proposed rIGA method delivers a speed-up factor proportional to p 2 . In a 2D mesh with four million elements and p = 5, the linear system resulting from rIGA is solved 22 times faster than the one from highly continuous IGA. In a 3D mesh with one million elements and p = 3, the linear rIGA system is solved 15 times faster than the IGA one.

We simulate electromagnetic (EM) measurements acquired with a logging-whiledrilling (LWD) instrument in a borehole environment. The measurements are used to assess electrical properties of rock formations. Logging instruments as well as rock formation properties are assumed to exhibit axial symmetry around the axis of a vertical borehole. The simulations are performed with a self-adaptive goal-oriented hp-finite element method that delivers exponential convergence rates in terms of the quantity of interest (for example, the difference in the electrical current measured at two receiver antennas) against the CPU time. Goal-oriented adaptivity allows for accurate approximations of the quantity of interest without the need to obtain an accurate solution in the entire computational domain. In particular, goal-oriented hp-adaptivity becomes essential to simulating LWD instruments, since it reduces the computational cost by several orders of magnitude with respect to the global energy-norm-based hp-adaptivity. Numerical results illustrate the efficiency and high accuracy of the method, and provide physical interpretation of resistivity measurements obtained with LWD instruments. These results also describe the advantages of using magnetic buffers in combination with solenoidal antennas for strengthening the measured EM signal so that the "signal-to-noise" ratio is minimized. Introduction.A plethora of energy-norm-based algorithms intended to generate optimal grids have been developed throughout recent decades (see, for example, [10, 23] and references therein) to accurately solve a large class of engineering problems. However, the energy-norm is a quantity of limited relevance for most engineering applications, especially when a particular objective is pursued, such as simulating the electromagnetic response of geophysical resistivity logging instruments in a borehole environment. In these instruments, the amplitude of the measurement (for example, the electric field) is typically several orders of magnitude smaller at the receiver antennas than at the transmitter antennas. Thus, small relative errors of the solution in the energy-norm do not imply small relative errors of the solution at the receiver

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In this paper we present computational cost estimates for parallel shared memory isogeometric multi-frontal solver. The estimates show that the ideal isogeometric shared memory parallel direct solver scales as O(p 2 log(N/p)) for one dimensional problems, O(Np 2 ) for two dimensional problems, and O(N 4/3 p 2 ) for three dimensional problems, where N is the number of degrees of freedom, and p is the polynomial order of approximation. The computational costs of the shared memory parallel isogeometric direct solver are compared with those corresponding to the sequential isogeometric direct solver, being the latest equal to O(Np 2 ) for the one dimensional case, O(N 1.5 p 3 ) for the two dimensional case, and O(N 2 p 3 ) for the three dimensional case. The shared memory version significantly reduces both the scalability in terms of N and p. Theoretical estimates are compared with numerical experiments performed with linear, quadratic, cubic, quartic, and quintic B-splines, in one and two spatial dimensions.Keywords: isogeometric finite element method, multi-frontal direct solver, computational cost, NVIDIA CUDA GPU Preprint submitted to Computers & Mathematics with ApplicationsMarch 27, 20141. Introduction Classical higher order finite element methods (FEM) [17,18] maintain only C 0 -continuity at the element interfaces, while isogeometric analysis (IGA) utilizes B-splines as basis functions, and thus, it delivers C k global continuity [14]. The higher continuity obtained across elements allows IGA to attain optimal convergence rates for any polynomial order, while using fewer degrees of freedom [3,1]. Nevertheless, this reduced count in the number of degrees of freedom may not immediately correlate with a computational cost reduction, since solution time per degree of freedom augments as the continuity is increased [10,13]. In spite of the increased cost of highercontinuous spaces, they have proven very popular and useful. For example, higher-continuous spaces have allowed the solution of higher-order partial di↵erential equations with elegance [7,28,29,51,16,15] as well as several non-linear problems of engineering interest [31,6,30,5,20,11,9,4]. Thus, e cient multi-frontal solvers for higher-continuous spaces are important.The multi-frontal solver is one of the state-of-the art algorithm for solving linear systems of equations [22,26]. It is a generalization of the frontal solver algorithm proposed in [33,21]. The multi-frontal algorithm constructs an assembly tree based on the analysis of the connectivity data or the geometry of the computational mesh. Finite elements are joint into pairs and fully assembled unknowns are eliminated within frontal matrices associated to multiple branches of the tree. The process is repeated until the root of the assembly tree is reached. Finally, the common interface problem is solved and partial backward substitutions are recursively called on the assembly tree.There exist parallel versions of the multi-frontal direct solver algorithm targeting distributed-memory, share...

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The paper presents a description of par3Dhp -a 3D, parallel, fully automatic hp-adaptive finite element code for elliptic and Maxwell problems. The parallel implementation is an extension of the sequential code 3Dhp90, which generates, in a fully automatic mode, optimal hp meshes for various boundary value problems. The system constitutes an infrastructure for a class of parallel hp adaptive computations. Its modular structure allows for an independent parallelization of each component of the system. The presented work addresses parallelization of these components, including distributed data structures, load balancing and domain redistribution, parallel (multi-frontal) solver, optimal hp mesh refinements, and a main control module. All components communicate through a distributed data structure, and the control module synchronizes work of all components. The concept of ghost elements has been used to simplify the communication algorithms for parallel mesh refinements. The system has been implemented in Fortran 90 and MPI, and the load balancing is done through an interface with the ZOLTAN library. Numerical results are presented for the model Fichera problem.Key words: Automatic hp-adaptivity, Finite Element Method, Parallel algorithms, High performance computing AcknowledgmentThe work of the second author has been supported by Air Force under Contract F49620-98-1-0255. The computations reported in this work were done through the National Science Foundation's National Partnership for Advanced Computational Infrastructure. The authors are greatly indebted to Jason Kurtz for numerous discussions on the subject. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) SPONSOR/MONITOR'S REPORT NUMBER(S) DISTRIBUTION/AVAILABILITY STATEMENTApproved for public release; distribution unlimited SUPPLEMENTARY NOTESThe original document contains color images. ABSTRACTThe paper presents a description of par3Dhp -a 3D, parallel, fully automatic hp-adaptive finite element code for elliptic and Maxwell problems. The parallel implementation is an extension of the sequential code 3Dhp90, which generates, in a fully automatic mode, optimal hp meshes for various boundary value problems. The system constitutes an infrastructure for a class of parallel hp adaptive computations. Its modular structure allows for an independent parallelization of each component of the system. The presented work addresses parallelization of these components, including distributed data structures, load balancing and domain redistribution, parallel (multi-frontal) solver, optimal hp mesh refinements, and a main control module. All components communicate through a distributed data structure, and the control module synchronizes work of all components. The concept of ghost elements has been used to simplify the communication algorithms for parallel mesh refinements. The system has been implemented in Fortran 90 and MPI, and the load balancing is done through an interface with the ZOLTAN library. Numerical results are pres...