Div First-Order System LL* (FOSLL*) for Second-Order Elliptic Partial Differential Equations

The first-order system LL* (FOSLL*) approach for general second-order elliptic partial differential equations was proposed and analyzed in [10], in order to retain the full efficiency of the L2 norm first-order system least-squares (FOSLS) ap- proach while exhibiting the generality of the inverse-norm FOSLS approach. The FOSLL* approach in [10] was applied to the div-curl system with added slack vari- ables, and hence it is quite complicated. In this paper, we apply the FOSLL* approach to the div system and establish its well-posedness. For the corresponding finite ele- ment approximation, we obtain a quasi-optimal a priori error bound under the same regularity assumption as the standard Galerkin method, but without the restriction to sufficiently small mesh size. Unlike the FOSLS approach, the FOSLL* approach does not have a free a posteriori error estimator, we then propose an explicit residual error estimator and establish its reliability and efficiency bound

Novel design for transparent high-pressure fuel injector nozzles

The efficiency and emissions of internal combustion (IC) engines are closely tied to the formation of the combustible air-fuel mixture. Direct-injection engines have become more common due to their increased practical flexibility and efficiency, and sprays dominate mixture formation in these engines. Spray formation, or rather the transition from a cylindrical liquid jet to a field of isolated droplets, is not completely understood. However, it is known that nozzle orifice flow and cavitation have an important effect on the formation of fuel injector sprays, even if the exact details of this effect remain unknown. A number of studies in recent years have used injectors with optically transparent nozzles (OTN) to allow observation of the nozzle orifice flow. Our goal in this work is to design various OTN concepts that mimic the flow inside commercial injector nozzles, at realistic fuel pressures, and yet still allow access to the very near nozzle region of the spray so that interior flow structure can be correlated with primary breakup dynamics. This goal has not been achieved until now because interior structures can be very complex, and the most appropriate optical materials are brittle and easily fractured by realistic fuel pressures. An OTN design that achieves realistic injection pressures and grants visual access to the interior flow and spray formation will be explained in detail. The design uses an acrylic nozzle, which is ideal for imaging the interior flow. This nozzle is supported from the outside with sapphire clamps, which reduces tensile stresses in the nozzle and increases the nozzle\u27s injection pressure capacity. An ensemble of nozzles were mechanically tested to prove this design concept

A PETSc parallel-in-time solver based on MGRIT algorithm

We address the development of a modular implementation of the MGRIT (MultiGrid-In-Time) algorithm to solve linear and nonlinear systems that arise from the discretization of evolutionary models with a parallel-in-time approach in the context of the PETSc (the Portable, Extensible Toolkit for Scientific computing) library. Our aim is to give the opportunity of predicting the performance gain achievable when using the MGRIT approach instead of the Time Stepping integrator (TS). To this end, we analyze the performance parameters of the algorithm that provide a-priori the best number of processing elements and grid levels to use to address the scaling of MGRIT, regarded as a parallel iterative algorithm proceeding along the time dimensio

Multilevel convergence analysis of multigrid-reduction-in-time

This paper presents a multilevel convergence framework for multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V- and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time stepping operator, analyzed directly and bounded in norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time stepping operator as parameters. The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge-Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge-Kutta schemes.Comment: 26 pages; 17 pages Supplementary Material