Stabilisation of spectral/hp element methods through spectral vanishing viscosity: Application to fluid mechanics modelling

Stabilisation of spectral/hp element methods through spectral vanishing viscosity: Application to fluid mechanics modellin

Interpolation error bounds for curvilinear finite elements and their implications on adaptive mesh refinement

This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.Mesh generation and adaptive renement are largely driven by the objective of minimizing the bounds on the interpolation error of the solution of the partial di erential equation (PDE) being solved. Thus, the characterization and analysis of interpolation error bounds for curved, high-order nite elements is often desired to e ciently obtain the solution of PDEs when using the nite element method (FEM). Although the order of convergence of the projection error in L2 is known for both straight-sided and curved-elements [1], an L1 estimate as used when studying interpolation errors is not available. Using a Taylor series expansion approach, we derive an interpolation error bound for both straight-sided and curved, high-order elements. The availability of this bound facilitates better node placement for minimizing interpolation error compared to the traditional approach of minimizing the Lebesgue constant as a proxy for interpolation error. This is useful for adaptation of the mesh in regions where increased resolution is needed and where the geometric curvature of the elements is high, e.g, boundary layer meshes. Our numerical experiments indicate that the error bounds derived using our technique are asymptotically similar to the actual error, i.e., if our interpolation error bound for an element is larger than it is for other elements, the actual error is also larger than it is for other elements. This type of bound not only provides an indicator for which curved elements to re ne but also suggests whether one should use traditional h-re nement or should modify the mapping function used to de ne elemental curvature. We have validated our bounds through a series of numerical experiments on both straight-sided and curved elements, and we report a summary of these results.The first author acknowledges support from the EU Horizon 2020 project ExaFLOW( grant 671571) and the PRISM project under EPSRC grant EP/L000407/1. The work of the second author was supported in part by the NIH/NIGMS Center for Integrative Biomedical Computing grant 2P41 RR0112553-12 and DOE NET DE-EE0004449 grant. The work of the third author was supported in part by the DOE NET DE-EE0004449 grant and ARO W911NF1210375 (Program Manager: Dr. Mike Coyle) grant

System Evolution, Feedback and Compliant Architectures

Proceedings, International Workshop on Feedback and Evolution in Software and Business Processes (FEAST 2000), Imperial College, London. Supported by EPSRCPostprintNon peer reviewe

Instances and connectors : issues for a second generation process language

This work is supported by UK EPSRC grants GR/L34433 and GR/L32699Over the past decade a variety of process languages have been defined, used and evaluated. It is now possible to consider second generation languages based on this experience. Rather than develop a second generation wish list this position paper explores two issues: instances and connectors. Instances relate to the relationship between a process model as a description and the, possibly multiple, enacting instances which are created from it. Connectors refers to the issue of concurrency control and achieving a higher level of abstraction in how parts of a model interact. We believe that these issues are key to developing systems which can effectively support business processes, and that they have not received sufficient attention within the process modelling community. Through exploring these issues we also illustrate our approach to designing a second generation process language.Postprin

From h to p efficiently: optimal implementation strategies for explicit time-dependent problems using the spectral/hp element method

We investigate the relative performance of a second-order Adams–Bashforth scheme and second-order and fourth-order Runge–Kutta schemes when time stepping a 2D linear advection problem discretised using a spectral/hp element technique for a range of different mesh sizes and polynomial orders. Numerical experiments explore the effects of short (two wavelengths) and long (32 wavelengths) time integration for sets of uniform and non-uniform meshes. The choice of time-integration scheme and discretisation together fixes a CFL limit that imposes a restriction on the maximum time step, which can be taken to ensure numerical stability. The number of steps, together with the order of the scheme, affects not only the runtime but also the accuracy of the solution. Through numerical experiments, we systematically highlight the relative effects of spatial resolution and choice of time integration on performance and provide general guidelines on how best to achieve the minimal execution time in order to obtain a prescribed solution accuracy. The significant role played by higher polynomial orders in reducing CPU time while preserving accuracy becomes more evident, especially for uniform meshes, compared with what has been typically considered when studying this type of problem.© 2014. The Authors. International Journal for Numerical Methods in Fluids published by John Wiley & Sons, Ltd

Optimising the performance of the spectral/hp element method with collective linear algebra operations

This is the final version of the article. Available from Elsevier via the DOI in this record.As computing hardware evolves, increasing core counts mean that memory bandwidth is becoming the deciding factor in attaining peak performance of numerical methods. High-order finite element methods, such as those implemented in the spectral/hp framework Nektar++, are particularly well-suited to this environment. Unlike low-order methods that typically utilise sparse storage, matrices representing high-order operators have greater density and richer structure. In this paper, we show how these qualities can be exploited to increase runtime performance on nodes that comprise a typical high-performance computing system, by amalgamating the action of key operators on multiple elements into a single, memory-efficient block. We investigate different strategies for achieving optimal performance across a range of polynomial orders and element types. As these strategies all depend on external factors such as BLAS implementation and the geometry of interest, we present a technique for automatically selecting the most efficient strategy at runtime.We thank D. Ekelschot and M. Turner for their assistance in generating the mesh and parameters for the simulation of Section 6. We also thank F. Witherden for initial discussions motivating this study. This work was funded in part by support from the libHPC II EPSRC project under grant EP/K038788/1. DM additionally acknowledges support under the Laminar Flow Control Centre funded by Airbus/EADS and EPSRC under grant EP/I037946. SJS acknowledges Royal Academy of Engineering support under their research chair scheme. We thank the Imperial College High Performance Computing Service for computing time used to calculate the results seen in Section 6. We additionally acknowledge access to ARCHER with support from the UK Turbulence Consortium under EPSRC grant EP/L000261/1

Combined CG-HDG Method for Elliptic Problems: Performance Model

We combine continuous and discontinuous Galerkin methods in the setting of a model diffusion problem. Starting from a hybrid discontinuous formulation, we replace element interiors by more general subsets of the computational domain - groups of elements that support a piecewise-polynomial continuous expansion. This step allows us to identify a~new weak formulation of Dirichlet boundary condition in the continuous framework. We examine the expected performance of a Galerkin solver that would use continuous Galerkin method with weak Dirichlet boundary conditions in each mesh partition and connect partitions weakly using trace variable as in HDG method