An optimal linear solver for the Jacobian system of the extreme type-II Ginzburg--Landau problem

This paper considers the extreme type-II Ginzburg--Landau equations, a nonlinear PDE model for describing the states of a wide range of superconductors. Based on properties of the Jacobian operator and an AMG strategy, a preconditioned Newton--Krylov method is constructed. After a finite-volume-type discretization, numerical experiments are done for representative two- and three-dimensional domains. Strong numerical evidence is provided that the number of Krylov iterations is independent of the dimension $n$ of the solution space, yielding an overall solver complexity of O(n)

A new level-dependent coarsegrid correction scheme for indefinite Helmholtz problems

In this paper we construct and analyse a level-dependent coarsegrid correction scheme for indefinite Helmholtz problems. This adapted multigrid method is capable of solving the Helmholtz equation on the finest grid using a series of multigrid cycles with a grid-dependent complex shift, leading to a stable correction scheme on all levels. It is rigourously shown that the adaptation of the complex shift throughout the multigrid cycle maintains the functionality of the two-grid correction scheme, as no smooth modes are amplified in or added to the error. In addition, a sufficiently smoothing relaxation scheme should be applied to ensure damping of the oscillatory error components. Numerical experiments on various benchmark problems show the method to be competitive with or even outperform the current state-of-the-art multigrid-preconditioned Krylov methods, like e.g. CSL-preconditioned GMRES or BiCGStab.Comment: 21 page

A matrix-free parallel solution method for the three-dimensional heterogeneous Helmholtz equation

The Helmholtz equation is related to seismic exploration, sonar, antennas, and medical imaging applications. It is one of the most challenging problems to solve in terms of accuracy and convergence due to the scalability issues of the numerical solvers. For 3D large-scale applications, high-performance parallel solvers are also needed. In this paper, a matrix-free parallel iterative solver is presented for the three-dimensional (3D) heterogeneous Helmholtz equation. We consider the preconditioned Krylov subspace methods for solving the linear system obtained from finite-difference discretization. The Complex Shifted Laplace Preconditioner (CSLP) is employed since it results in a linear increase in the number of iterations as a function of the wavenumber. The preconditioner is approximately inverted using one parallel 3D multigrid cycle. For parallel computing, the global domain is partitioned blockwise. The matrix-vector multiplication and preconditioning operator are implemented in a matrix-free way instead of constructing large, memory-consuming coefficient matrices. Numerical experiments of 3D model problems demonstrate the robustness and outstanding strong scaling of our matrix-free parallel solution method. Moreover, the weak parallel scalability indicates our approach is suitable for realistic 3D heterogeneous Helmholtz problems with minimized pollution error.Comment: 25 pages, 15 figures, manuscript submitted to a special issue of conference NMLSP202

Robust and scalable 3-D geo-electromagnetic modelling approach using the finite element method

We present a robust and scalable solver for time-harmonic Maxwell's equations for problems with large conductivity contrasts, wide range of frequencies, stretched grids and locally refined meshes. The solver is part of the fully distributed adaptive 3-D electromagnetic modelling scheme which employs the finite element method and unstructured non-conforming hexahedral meshes for spatial discretization using the open-source software deal.II. We use the complex-valued electric field formulation and split it into two real-valued equations for which we utilize an optimal block-diagonal pre-conditioner. Application of this pre-conditioner requires the solution of two smaller real-valued symmetric problems. We solve them by using either a direct solver or the conjugate gradient method pre-conditioned with the recently introduced auxiliary space technique. The auxiliary space pre-conditioner reformulates the original problem in form of several simpler ones, which are then solved using highly efficient algebraic multigrid methods. In this paper, we consider the magnetotelluric case and verify our numerical scheme by using COMMEMI 3-D models. Afterwards, we run a series of numerical experiments and demonstrate that the solver converges in a small number of iterations for a wide frequency range and variable problem sizes. The number of iterations is independent of the problem size, but exhibits a mild dependency on frequency. To test the stability of the method on locally refined meshes, we have implemented a residual-based a posteriori error estimator and compared it with uniform mesh refinement for problems up to 200 million unknowns. We test the scalability of the most time consuming parts of our code and show that they fulfill the strong scaling assumption as long as each MPI process possesses enough degrees of freedom to alleviate communication overburden. Finally, we refer back to a direct solver-based pre-conditioner and analyse its complexity in time. The results show that for multiple right-hand sides the direct solver-based pre-conditioner can still be faster for problems of medium size. On the other hand, it also shows non-linear growth in memory, whereas the auxiliary space method increases only linearly.ISSN:0956-540XISSN:1365-246

Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper

Numerical Methods for Simulating Multiphase Electrohydrodynamic Flows with Application to Liquid Fuel Injection

One approach to small-scale fuel injection is to capitalize upon the benefits of electrohydrodynamics (EHD) and enhance fuel atomization. There are many potential advantages to EHD aided atomization for combustion, such as smaller droplets, wider spray cone, and the ability to control and tune the spray for improved performance. Electrohydrodynamic flows and sprays have drawn increasing interest in recent years, yet key questions regarding the complex interactions among electrostatic charge, electric fields, and the dynamics of atomizing liquids remain unanswered. The complex, multi-physics and multi-scale nature of EHD atomization processes limits both experimental and computational explorations. In this work, novel, numerically sharp methods are developed and subsequently employed in high-fidelity direct numerical simulations of electrically charged liquid hydrocarbon jets. The level set approach is combined with the ghost fluid method (GFM) to accurately simulate primary atomization phenomena for this class of flows. Surface effects at the phase interface as well as bulk dynamics are modeled in an accurate and robust manner. The new methods are implemented within a conservative finite difference scheme of high-order accuracy that employs state-of-the-art interface transport techniques. This approach, validated using several cases with exact analytic solutions, demonstrates significant improvements in accuracy and efficiency compared to previous methods used for EHD simulations. As a final validation, the computational scheme is applied in direct numerical simulation of a charged and uncharged liquid kerosene jet. Then, a detailed numerical study of EHD atomization is conducted for a range of relevant dimensionless parameters to predict the onset of liquid break-up, identify characteristic modes of liquid disintegration, and report elucidating statistics such as drop size and spray dispersion. Because the methodologies developed and validated in this work open new, simulations-based avenues of exploration within a broader category of electrohydrodynamics, some perspectives on extensions or continuations of this work are offered in conclusion

Relaxation-Corrected Bootstrap Algebraic Multigrid (rBAMG)

Bootstrap Algebraic Multigrid (BAMG) is a multigrid-based solver for matrix equations of the form Ax = b. Its aim is to automatically determine the interpolation weights used in algebraic multigrid (AMG) by locally fitting a set of test vectors that have been relaxed as solutions to the corresponding homogeneous equation, Ax = 0. This thesis introduces an indirect operator based interpolation scheme for BAMG that determines the interpolation weights indirectly by “collapsing” the unwanted connections in “operator interpolation”. Compared to BAMG, this indirect BAMG approach (iBAMG) is more in the spirit of classical AMG, which collapses unwanted connections in operator interpolation based on the (restrictive) assumption that smooth error is locally constant. This thesis also develops another form of BAMG, called rBAMG, that involves modifying the least-squares process by temporarily relaxing on the test vectors at the fine-grid interpolation points. The theory here shows that, under fairly general conditions, iBAMG and rBAMG are equivalent. Simplicity and potentially greater generality favor rBAMG, so this algorithm is at the focus of the numerical performance study here. The rBAMG setup process involves several components that are developed in this thesis. Besides the new least-squares principle involving the residuals of the test vectors, a simple extrapolation scheme is developed to accurately estimate the convergence factors of the evolving AMG solver. Such a capability is essential to effective development of a fast solver, and the approach introduced here proves to be much more effective than the conventional approach of just observing successive error reduction factors. Another component of the setup process is the use of the current V-cycle to ensure its effectiveness or, when poor convergence is observed, to expose error components that are not being properly attenuated. How we coordinate use of these evolving error components together with the original test vectors to direct the setup process is a critical issue to rBAMG’s effectiveness. Another related component is the scaling and recombination Ritz process that targets the so-called weak approximation property in an attempt to reveal the important elements of these evolving error and test vector spaces. The details of the components used here are spelled out in what follows. The study of rBAMG here is an attempt to systematically analyze the behavior of the algorithm in terms relative to several parameters. The focus here is on the number of test vectors, the number of relaxations applied to them, and the dimension of the matrix to which the scheme is applied. A large number of other parameters and options could also be considered, including different cycling strategies, other coarsening strategies (e. g., computing several eigenvector approximations on coarse levels), different numbers of relaxation sweeps on coarse levels, different possible strategies for combining test vectors and error components produced by the current cycles, and so on. Studying all of these options and parameters would not be feasible here. Instead, reasonable choices are made based on some sample studies (that, in the interest of space, we choose not to document here), with the hope that the rBAMG algorithm studied here is generally fairly effective and robust. Our analysis is thus able to focus on how this scheme behaves numerically in the face of increasing the numbers of test vectors and relaxation sweeps performed on them, as well as the problem sizes