Potential-based Formulations of the Navier-Stokes Equations and their Application

Based on a Clebsch-like velocity representation and a combination of classical variational principles for the special cases of ideal and Stokes flow a novel discontinuous Lagrangian is constructed; it bypasses the known problems associated with non-physical solutions and recovers the classical Navier-Stokes equations together with the balance of inner energy in the limit when an emerging characteristic frequency parameter tends to infinity. Additionally, a generalized Clebsch transformation for viscous flow is established for the first time. Next, an exact first integral of the unsteady, three-dimensional, incompressible Navier-Stokes equations is derived; following which gauge freedoms are explored leading to favourable reductions in the complexity of the equation set and number of unknowns, enabling a self-adjoint variational principle for steady viscous flow to be constructed. Concurrently, appropriate commonly occurring physical and auxiliary boundary conditions are prescribed, including establishment of a first integral for the dynamic boundary condition at a free surface. Starting from this new formulation, three classical flow problems are considered, the results obtained being in total agreement with solutions in the open literature. A new least-squares finite element method based on the first integral of the steady two-dimensional, incompressible, Navier-Stokes equations is developed, with optimal convergence rates established theoretically. The method is analysed comprehensively, thoroughly validated and shown to be competitive when compared to a corresponding, standard, primitive-variable, finite element formulation. Implementation details are provided, and the well-known problem of mass conservation addressed and resolved via selective weighting. The attractive positive definiteness of the resulting linear systems enables employment of a customized scalable algebraic multigrid method for efficient error reduction. The solution of several engineering related problems from the fields of lubrication and film flow demonstrate the flexibility and efficiency of the proposed method, including the case of unsteady flow, while revealing new physical insights of interest in their own right

Schnelle Löser für partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Global convection in Earth's mantle : advanced numerical methods and extreme-scale simulations

The thermal convection of rock in Earth's mantle and associated plate tectonics are modeled by nonlinear incompressible Stokes and energy equations. This dissertation focuses on the development of advanced, scalable linear and nonlinear solvers for numerical simulations of realistic instantaneous mantle flow, where we must overcome several computational challenges. The most notable challenges are the severe nonlinearity, heterogeneity, and anisotropy due to the mantle's rheology as well as a wide range of spatial scales and highly localized features. Resolving the crucial small scale features efficiently necessitates adaptive methods, while computational results greatly benefit from a high accuracy per degree of freedom and local mass conservation. Consequently, the discretization of Earth's mantle is carried out by high-order finite elements on aggressively adaptively refined hexahedral meshes with a continuous, nodal velocity approximation and a discontinuous, modal pressure approximation. These velocity--pressure pairings yield optimal asymptotic convergence rates of the finite element approximation to the infinite-dimensional solution with decreasing mesh element size, are inf-sup stable on general, non-conforming hexahedral meshes with "hanging nodes,'' and have the advantage of preserving mass locally at the element level due to the discontinuous pressure. However, because of the difficulties cited above and the desired accuracy, the large implicit systems to be solved are extremely poorly conditioned and sophisticated linear and nonlinear solvers including powerful preconditioning techniques are required. The nonlinear Stokes system is solved using a grid continuation, inexact Newton--Krylov method. We measure the residual of the momentum equation in the H⁻¹-norm for backtracking line search to avoid overly conservative update steps that are significantly reduced from one. The Newton linearization is augmented by a perturbation of a highly nonlinear term in mantle's rheology, resulting in dramatically improved nonlinear convergence. We present a new Schur complement-based Stokes preconditioner, weighted BFBT, that exhibits robust fast convergence for Stokes problems with smooth but highly varying (up to 10 orders of magnitude) viscosities, optimal algorithmic scalability with respect to mesh refinement, and only a mild dependence on the polynomial order of high-order finite element discretizations. In addition, we derive theoretical eigenvalue bounds to prove spectral equivalence of our inverse Schur complement approximation. Finally, we present a parallel hybrid spectral--geometric--algebraic multigrid (HMG) to approximate the inverses of the Stokes system's viscous block and variable-coefficient pressure Poisson operators within weighted BFBT. Building on the parallel scalability of HMG, our Stokes solver demonstrates excellent parallel scalability to 1.6 million CPU cores without sacrificing algorithmic optimality.Computational Science, Engineering, and Mathematic

Schnelle Löser für Partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Operator-adapted wavelets for finite-element differential forms

We introduce in this paper an operator-adapted multiresolution analysis for finite-element differential forms. From a given continuous, linear, bijective, and self-adjoint positive-definite operator L, a hierarchy of basis functions and associated wavelets for discrete differential forms is constructed in a fine-to-coarse fashion and in quasilinear time. The resulting wavelets are L-orthogonal across all scales, and can be used to derive a Galerkin discretization of the operator such that its stiffness matrix becomes block-diagonal, with uniformly well-conditioned and sparse blocks. Because our approach applies to arbitrary differential p-forms, we can derive both scalar-valued and vector-valued wavelets block-diagonalizing a prescribed operator. We also discuss the generality of the construction by pointing out that it applies to various types of computational grids, offers arbitrary smoothness orders of basis functions and wavelets, and can accommodate linear differential constraints such as divergence-freeness. Finally, we demonstrate the benefits of the corresponding operator-adapted multiresolution decomposition for coarse-graining and model reduction of linear and non-linear partial differential equations

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Asynchronous Stabilisation and Assembly Techniques for Additive Multigrid

Multigrid solvers are among the best solvers in the world, but once applied in the real world there are issues they must overcome. Many multigrid phases exhibit low concurrency. Mesh and matrix assembly are challenging to parallelise and introduce algorithmic latency. Dynamically adaptive codes exacerbate these issues. Multigrid codes require the computation of a cascade of matrices and dynamic adaptivity means these matrices are recomputed throughout the solve. Existing methods to compute the matrices are expensive and delay the solve. Non- trivial material parameters further increase the cost of accurate equation integration. We propose to assemble all matrix equations as stencils in a delayed element-wise fashion. Early multigrid iterations use cheap geometric approximations and more accurate updated stencil integrations are computed in parallel with the multigrid cycles. New stencil integrations are evaluated lazily and asynchronously fed to the solver once they become available. They do not delay multigrid iterations. We deploy stencil integrations as parallel tasks that are picked up by cores that would otherwise be idle. Coarse grid solves in multiplicative multigrid also exhibit limited concurrency. Small coarse mesh sizes correspond to small computational workload and require costly synchronisation steps. This acts as a bottleneck and delays solver iterations. Additive multigrid avoids this restriction, but becomes unstable for non-trivial material parameters as additive coarse grid levels tend to overcorrect. This leads to oscillations. We propose a new additive variant, adAFAC-x, with a stabilisation parameter that damps coarse grid corrections to remove oscillations. Per-level we solve an additional equation that produces an auxiliary correction. The auxiliary correction can be computed additively to the rest of the solve and uses ideas similar to smoothed aggregation multigrid to anticipate overcorrections. Pipelining techniques allow adAFAC-x to be written using single-touch semantics on a dynamically adaptive mesh