Browsing by Subject "Poisson equation"
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Item Convergence Order Studies for Elliptic Test Problems with COMSOL Multiphysics(2008) Yang, Shiming; Gobbert, Matthias K.The convergence order of fi nite elements is related to the polynomial order of the basis functions used on each element, with higher order polynomials yielding better convergence orders. However, two issues can prevent this convergence order from being achieved: the poor approximation of curved boundaries by polygonal meshes and lack of regularity of the PDE solution. We show studies for Lagrange elements of degrees 1 through 5 applied to the classical test problem of the Poisson equation with Dirichlet boundary condition. We consider this problem in 1, 2, and 3 spatial dimensions and on domains with polygonal and with curved boundaries. The observed convergence orders in the norm of the error between FEM and PDE solution demonstrate that they are limited by the regularity of the solution and are degraded signifi cantly on domains with non-polygonal boundaries. All numerical tests are carried out with COMSOL Multiphysics.Item FEM Convergence for PDEs with Point Sources in 2-D and 3-D(2015) Gobbert, M.; Kalayeh, K. M.; Graf, J. S.Numerical theory provides the basis for quantification of the accuracy and reliability of a FEM solution by error estimates on the FEM error vs. the mesh spacing of the FEM mesh. This paper presents techniques needed in COMSOL 5.1 to perform computational studies for elliptic test problems in two and three space dimensions that demonstrate this theory by computing the convergence order of the FEM error. In particular, we show how to perform these techniques for a problem involving a point source modeled by a Dirac delta distribution as forcing term. This demonstrates that PDE problems with a non-smooth source term necessarily have degraded convergence order compared to problems with smooth right-hand sides and thus can be most efficiently solved by low-order FEM such as linear Lagrange elements.Item FEM Convergence Studies for 2-D and 3-D Elliptic PDEs with Smooth and Non-Smooth Source Terms in COMSOL 5.1(2015) Kalayeh, Kourosh M.; Graf, Jonathan S.; Gobbert, Matthias K.Numerical theory provides the basis for quanti cation on the accuracy and reliability of a FEM solution by error estimates on the FEM error vs. the mesh spacing of the FEM mesh. This paper presents techniques needed in COMSOL 5.1 to perform computational studies for an elliptic test problem in two and three space dimensions that demonstrate this theory by computing the convergence order of the FEM error. For a PDE with smooth right-hand side, linear Lagrange nite elements exhibit second order convergence for all space dimensions. We also show how to perform these techniques for a problem involving a point source modeled by a Dirac delta distribution as forcing term. This demonstrates that PDE problems with a non-smooth source term necessarily have degraded convergence order and thus can be most e ciently solved by low-order FEM such as linear Lagrange elements. Detailed instructions for obtaining the results are included in an appendix.Item Finite Element Convergence Studies Using COMSOL 4.0a and LiveLink for MATLAB(2010) Trott, David W.; Gobbert, Matthias K.In order to gauge how reasonable a finite element solution to a partial differential equation is on a given mesh, a common strategy is to refine the mesh, compute the solution on the finer mesh, and use the solutions on the two meshes for a qualitative comparison. The theory of the finite element method (FEM) makes these comparisons quantitative by estimating the convergence order of the FEM error on a sequence of progressively finer meshes obtained by uniform mesh refinement. We show in detail how to carry out convergence studies of this type using the graphical user interface (GUI) of COMSOL 4.0a as well as using COMSOL’s LiveLink for MATLAB on the example of Lagrange elements of varying polynomial degrees. Conducting the convergence study in this manner shows how to quantify the convergence of FEM solutions and brings out the potential benefit of using higher order elements. The interconnection of COMSOL with MATLAB allows for a convenient automization of the study that is not possible through the use of the GUI alone, but is vital for reproducible studies and useful for running studies in batch mode on computing clusters.Item A GPU Memory System Comparison for an Elliptic Test ProblemWang, Yu; Gobbert, Matthias K.; Olano, Marc; Griffin, WesleyThis paper presents GPU-based solutions to the Poisson equation with homogeneous Dirichlet boundary conditions in two spatial dimensions. This problem has well-understood behavior, but similar computation to many more complex real-world problems. We analyze the GPU performance using three types of memory access in the CUDA memory model (direct access to global memory, texture access, and shared memory). Based on data locality, different CUDA algorithms are designed to accommodate the different device memory performance behaviors. We present a performance study on the speedup of our GPU-based solutions on an NVIDIA Tesla C2070 over serial code. By relating the data access pattern and its spatial locality, our results show that an algorithm using global memory with coalesced reads outperforms the other memory systems and allows effective solvers using single precision floating points.Item The Optimal Relaxation Parameter for the SOR Method Applied to the Poisson Equation in Any Space Dimensions(Elsevier Ltd, 2009-03-24) Yang, Shiming; Gobbert, Matthias K.The finite difference discretization of the Poisson equation with Dirichlet boundary conditions leads to a large, sparse system of linear equations for the solution values at the interior mesh points. This problem is a popular and useful model problem for performance comparisons of iterative methods for the solution of linear systems. To use the successive overrelaxation (SOR) method in these comparisons, a formula for the optimal value of its relaxation parameter is needed. In standard texts, this value is only available for the case of two space dimensions, even though the model problem is also instructive in higher dimensions. This note extends the derivation of the optimal relaxation parameter to any space dimension and confirms its validity by means of test calculations in three dimensions.Item Parallel Performance Studies for an Elliptic Test Problem(2008) Gobbert, Matthias K.The performance of parallel computer code depends on an intricate interplay of the processors, the architecture of the compute nodes, their interconnect network, the numerical algorithm, and the scheduling policy used. The solution of large, sparse, highly structured systems of linear equations by an iterative linear solver that requires communication between the parallel processes at every iteration is an instructive test of this interplay. This note considers the classical elliptic test problem of a Poisson equation with Dirichlet boundary conditions, whose approximation by the finite difference method results in a linear system of this type. Our existing implementation of the conjugate gradient method for the iterative solution of this system is known to have the potential to perform well up to many parallel processes, provided the interconnect network has low latency. Since the algorithm is known to be memory bound, it is also vital for good performance that the architecture of the nodes in conjunction with the scheduling policy does not create a bottleneck. The results presented here show excellent performance the cluster hpc in the UMBC High Performance Computing Facility and give guidance on the scheduling policy to be implemented. Specifically, they confirm that it is beneficial to use all four cores of the two dual-core processors on each node simultaneously, giving us in effect a computer that can run jobs efficiently with up to 128 parallel processes.Item Performance comparison of Intel Xeon Phi Knights Landing(2017) Jabbie, Ishmail A.; Owen, George; Whiteley, Benjamin; Gobbert, Matthias K.The Intel Xeon Phi is a many-core processor with a theoretical peak performance of over 3 TFLOP/s of double precision. We contrast the performance of the second-generation Intel Xeon Phi, code-named Knights Landing (KNL), to the first-generation Intel Xeon Phi, codenamed Knights Corner (KNC), as well as to a node with two multi-core CPUs as baseline reference. The test code solves the classical elliptic test problem of the Poisson equation whose performance is prototypical for the computational kernel in many numerical methods for partial differential equations. The results show that the KNL can perform approximately four times faster than the KNC or than two CPUs, provided the problem fits into the 16 GB of on-chip MCDRAM memory of the KNL. The studies also confirm the nominal five times faster speed of the new high-performance MCDRAM memory in the KNL compared to the DDR4 memory of the node. We demonstrate the ease of porting code to the KNL by focusing on performance that was achieved by only re-compiling hybrid MPI+OpenMP code with a KNL flag.Item Performance Studies for Multithreading in Matlab with Usage Instructions on hpc(2009) Sharma, Neeraj; Gobbert, Matthias K.This report explores the use of multiple computational cores by multithreading in the software package Matlab on a compute node with two dual-core AMD Opteron processors. After testing the built-in functions of Matlab for a small test problem, we consider a classical test problem resulting from a finite difference discretization of the Poisson equation in two spatial dimensions. The results demonstrate that the use of more than one thread is often not very beneficial for Matlab code. This suggests that Matlab jobs should be limited to using one core by default to allow for the fastest throughput of the largest number of jobs. The comparison of solving the same problem with a code using the source code language C indicates that Matlab uses more memory and takes longer; this has to be contrasted with the productivity gains possible of programming in Matlab. This report also provides detailed information on how to run Matlab jobs in the UMBC High Performance Computing Facility.Item Pushing the Limits of the Maya Cluster(2014) Cunningham, Adam; Payton, Gerald; Slettebak, Jack; Pou, Jordi Wolfson; Graf, Jonathan; Huang, Xuan; Khuvis, Samuel; Gobbert, Matthias K.; Salter, Thomas; Mountain, David J.Parallelization of code, using multiple cores/threads, and heterogeneous computing, using the CPU with other devices, has come to the forefront of computing as methods to reduce the execution time of computationally demanding algorithms. For our project, we test various hardware setups on the maya cluster at UMBC, which include multiple nodes and GPUs, by solving the Poisson equation using the conjugate gradient method. To explore these different setups, we made use of both industry benchmarks and our own code, which we design using the compilers native to each device and API. We fi nd significant gains both in using a heterogeneous model and after parallelizing our code.Item Strong and Weak Scalability Studies for the 2-D Poisson Equation on the Taki 2018 Cluster(2019) Barajas, Carlos; Gobbert, Matthias K.The new 2018 nodes in the cluster taki in the UMBC High Performance Computing Facility contain two 18-core Intel Skylake CPUs and 384 GB of memory per node, connected by an EDR (Enhanced Data Rate) InfiniBand interconnect. Parallel performance studies for the memory-bound test problem of the Poisson equation in two spatial dimensions yield several conclusions for the operation of the CPU cluster in taki. Strong scalability studies demonstrate excellent performance when using multiple nodes due to the low latency of the high-performance interconnect and good speedup when using all cores of the multi-core CPUs. Weak scalability studies confirm that best throughput is achieved by using all cores on a shared-memory node. Comparisons to results on the 2009 and 2013 nodes bring out that core-per-core performance of serial code improvements have stalled, but that node-pernode performance of parallel code continues to improve due to the larger number of cores available on a node. These observations compel the recommendations that serial code should use the 2009 and 2013 nodes of taki and parallel code is needed to take full advantage of the high-memory 2018 nodes. Comparisons between several compilers and several implementations of the MPI standard justify the choice of the Intel suite as the default on taki.