Multi-Level Mixed Finite Element Methods Based on Different Iterations for the Steady Boussinesq Problem

In this paper, we propose a stabilized finite volume method based on the Newton's type iteration for the steady incompressible magnetohydrodynamics (MHD) problem. In order to reduce the computational complexity, the lowest order mixed finite element pair ( ‐ ‐ ) is adopted to approximate the velocity, pressure and magnetic fields, and the multiscale enrichment method is introduced to overcome the restriction of discrete inf‐sup (LBB) condition. We firstly solve the incompressible MHD equations by the Newton iterations on a coarse mesh with the mesh size , stability and convergence results of numerical solutions are provided. Secondly, the combination of the two‐level method with the Newton iteration is used to approximate the considered problem, a coupled nonlinear problem is solved on the coarse mesh , then the Stokes and Maxwell equations are considered on a fine mesh with the mesh size , the uniform stability and optimal error estimates of two‐level Newton iterative method are provided. Theoretical findings show that the two‐level method has the same order as the one‐level method in ‐norm as long as the mesh sizes satisfy . However, the two‐level method involves much less work than the one‐level method. Finally, some numerical examples are presented to demonstrate the effectiveness of the considered numerical schemes.

Stability and convergence of iterative methods related to viscosities for the 2D/3D steady Navier–Stokes equations

Engineering problems are often characterized by significant uncertainty in their material parameters. Multilevel sampling methods are a straightforward manner to account for this uncertainty. The most well known multilevel method is the Multilevel Monte Carlo method (MLMC). First developed by Giles, see [1], this method relies on a hierarchy of successive refined Finite Element meshes of the considered engineering problem, in order to achieve a computational speedup. Most of the samples are taken on coarse and computationally cheap meshes, while a decreasing number of samples are taken on finer and computationally expensive meshes. Classically, the mesh hierarchy is constructed by selecting a coarse mesh discretization of the problem, and recursively applying an h-refinement approach to it, see [2]. This will be referred to as h-MLMC. However, in the h-MLMC mesh hierarchy, the number of degrees of freedom increases almost geometrical with increasing level, leading to a large computational cost. An efficient manner to reduce this computational cost, is by means of the novel sampling method called p-refined Multilevel Quasi-Monte Carlo (p-MLQMC), see [3]. The p-MLQMC method uses a hierarchy of p-refined Finite Element meshes, combined with a deterministic Quasi-Monte Carlo sampling rule. This combination significantly reduced the computational cost with respect to h-MLMC. However, the p-MLQMC method presents the practitioner with a challenge. This challenge consists in adequately incorporating the uncertainty, represented as a random field, in the Finite Element model. In previous work, see [4], we have tackled this challenge by investigating how the evaluation points, used to calculate point evaluations of the random field by means of the Karhunen-Loève (KL) expansion, need to be selected in order to achieve the lowest computational cost. We found that using sets of nested evaluation points across the mesh hierarchy, i.e., the Local Nested Approach (LNA), yields a speedup up to a factor 5 with respect to sets consisting of non-nested evaluation points, i.e., the Non-Nested Approach (NNA). Furthermore, we have shown that p-MLQMC-LNA yields a speedup up to a factor 70 with respected to h-MLMC. Currently, our research focus lies on implementing the use of higher order Quasi-Monte Carlo rules, and hierarchical shape functions in p-MLQMC. Both paths show promising results for further computational savings in the p-MLQMC method. All the aforementioned implementations are benchmarked on a slope stability problem, with spatially varying uncertainty in the ground. The chosen quantity of interest (QoI) consists of the vertical displacement of the top of the slope.[1] Michael B. Giles. Multilevel Monte Carlo path simulation. Oper. Res., 56(3):607–617, 2008. [2] K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup. Multilevel Monte Carlo methods and applications to elliptic pdes with random coefficients. Comput. Vis. Sci., 14(1):3, Aug 2011. [3] Philippe Blondeel, Pieterjan Robbe, Cédric Van hoorickx, Stijn François, Geert Lombaert, and Stefan Vandewalle. p-refined multilevel quasi-monte carlo for galerkin finite element methods with applications in civil engineering. Algorithms, 13(5), 2020. [4] Philippe Blondeel, Pieterjan Robbe, Stijn François, Geert Lombaert, and Stefan Vandewalle. On the selection of random field evaluation points in the p-mlqmc method. arXiv, 2020.

CBS constants for multilevel splitting of graph-Laplacian and application to preconditioning of discontinuous Galerkin systems

We consider numerical simulation of blood flows in the artery using multilevel domain decomposition methods. Because of the complex geometry, the construction and the solve of the coarse problem take a large percentage of the total compute time in the multilevel method. In this paper, we introduce a one-dimensional central-line model of the blood flow and use its stabilized finite element discretization to construct a coarse preconditioner. With suitable restriction and extension operators, we obtain a two-level additive Schwarz preconditioner for two- and three-dimensional problems. We present some numerical experiments with different arteries to show the efficiency and robustness of the new coarse preconditioner whose computational cost is considerably lower than other coarse preconditioners constructed using the two- or three-dimensional geometry of the artery.

In this paper, we examine a number of additive and multiplicative multilevel iterative methods and preconditioners in the setting of two-dimensional local mesh refinement. While standard multilevel methods are effective for uniform refinement-based discretizations of elliptic equations, they tend to be less effective for algebraic systems, which arise from discretizations on locally refined meshes, losing their optimal behavior in both storage and computational complexity. Our primary focus here is on Bramble, Pasciak, and Xu (BPX)-style additive and multiplicative multilevel preconditioners, and on various stabilizations of the additive and multiplicative hierarchical basis (HB) method, and their use in the local mesh refinement setting. In parts I and II of this trilogy, it was shown that both BPX and wavelet stabilizations of HB have uniformly bounded condition numbers on several classes of locally refined two- and three-dimensional meshes based on fairly standard (and easily implementable) red and red-green mesh refinement algorithms. In this third part of the trilogy, we describe in detail the implementation of these types of algorithms, including detailed discussions of the data structures and traversal algorithms we employ for obtaining optimal storage and computational complexity in our implementations. We show how each of the algorithms can be implemented using standard data types, available in languages such as C and FORTRAN, so that the resulting algorithms have optimal (linear) storage requirements, and so that the resulting multilevel method or preconditioner can be applied with optimal (linear) computational costs. We have successfully used these data structure ideas for both MATLAB and C implementations using the FEtk, an open source finite element software package. We finish the paper with a sequence of numerical experiments illustrating the effectiveness of a number of BPX and stabilized HB variants for several examples requiring local refinement.

In this paper, we develop and design the two‐level finite element iterative methods for the stationary thermally coupled incompressible MHD equations. The considered numerical schemes are based on the classical iterations with one correction. First, under some strong uniqueness conditions, the rough solutions are obtained on the coarse mesh with the mesh size and the Simple, Oseen, and Newton iterations, respectively. Three kinds of corrections are made by solving a linear problem on the fine mesh with mesh size with different viscosities. Finally, under the weak uniqueness condition, the stationary thermally coupled incompressible MHD is solved by the one‐level finite element Oseen iteration on the fine mesh. The uniform stability and convergence of these two‐level iterative methods are analyzed with respect to the mesh sizes and iterative times . Extensive numerical results are presented to demonstrate the established theoretical findings and show the performances of these two‐level iterative schemes with different viscosities.

Stability and convergence of two-level iterative methods for the stationary incompressible magnetohydrodynamics with different Reynolds numbers

We develop a convergence theory for two level and multilevel additive Schwarz domain decomposition methods for elliptic and parabolic problems on general unstructured meshes in two and three dimensions. The coarse and fine grids are assumed only to be shape regular, and the domains formed by the coarse and fine grids need not be identical. In this general setting, our convergence theory leads to completely local bounds for the condition numbers of two level additive Schwarz methods, which imply that these condition numbers are optimal, or independent of fine and coarse mesh sizes and subdomain sizes if the overlap amount of a subdomain with its neighbors varies proportionally to the subdomain size. In particular, we will show that additive Schwarz algorithms are still very efficient for nonselfadjoint parabolic problems with only symmetric, positive definite solvers both for local subproblems and for the global coarse problem. These conclusions for elliptic and parabolic problems improve our earlier results in [12, 15, 16]. Finally, the convergence theory is applied to multilevel additive Schwarz algorithms. Under some very weak assumptions on the fine mesh and coarser meshes, e.g., no requirements on the relation between neighboring coarse level meshes, we are able to derive a condition number bound of the orderO(ρ2 L 2), whereρ = max1≤l≤L(h l +l− 1)/δ l,h l is the element size of thelth level mesh,δ l the overlap of subdomains on thelth level mesh, andL the number of mesh levels.

Many large nonlinear optimization problems are based upon a hierarchy of models, corresponding to levels of discretization or detail in the problem. Optimization-based multilevel methods – that is, multilevel methods based on solving coarser versions of an optimization problem – are designed to solve such multilevel problems efficiently by taking explicit advantage of the hierarchy of models. The methods are generalizations of multigrid methods for solving partial differential equations. These multilevel methods are a powerful tool, but they will not lead to improved performance over traditional algorithms for all optimization problems. We develop techniques whereby a particular multilevel method can assess the properties of the optimization problem, with the goal of automatically determining whether it is well suited for the multilevel algorithm. We also show that our diagnostic tests are sufficient to measure the properties of the optimization problem relevant to the performance of the multilevel method.

In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal differentiation and advection terms are treated by the characteristic scheme. Under some conditions about the coefficients and exact solution, optimal error estimates for the numerical solution are obtained. Furthermore, the twogrid characteristic finite volume methods involve solving a nonlinear equation on coarse mesh with mesh size H, a large linear problem for the Oseen two-grid characteristic finite volume method on a fine mesh with mesh size h = O(H 2 ) or a large linear problem for the Newton two-grid characteristic finite volume method on a fine mesh with mesh size h = O(|log h| 1/2 H 3 ). These methods we studied provide the same convergence rate as that of the characteristic finite volume method, which involves solving one large nonlinear problem on a fine mesh with mesh size h. Some numerical results are presented to demonstrate the efficiency of the proposed methods.

We explored how mesh size can affect the description of the macroinvertebrate community in large rivers using data from artificial substrate samplers in the main channel of the Savannah River near Augusta, Georgia, and from fine sediments in the main channel of the Upper Mississippi River near Cape Girardeau, Missouri. Samples from the Savannah River were collected on five occasions between 2000–2003 and processed through coarse-mesh (1.8 mm) and fine-mesh (0.5 mm) sieves. Samples from the Mississippi River were collected annually in 2002–2004 and processed through coarse-mesh (1.18 mm) and fine-mesh (0.355 mm) sieves. These mesh sizes contrast procedures associated with long-term studies (coarse mesh) and procedures that are frequently used or recommended now (fine mesh). In both rivers, coarse mesh greatly underestimated densities, capturing only 35% of the total macroinvertebrates in the Savannah River and 20% of the total numbers in the Mississippi River relative to the fine mesh. As a result, the density and relative abundance of dominant taxonomic groups differed between mesh sizes and among sampling dates. The differences for relative abundance, assemblage structure, and biometrics between the fine and coarse meshes were not consistent between rivers. Non-metric Multidimensional Scaling indicated in the Savannah River that the overall macroinvertebrate assemblage structure differed based on year and site but not mesh size, whereas assemblage structure in the Mississippi River differed based on mesh size. Similarly, biometrics from data with coarse and fine meshes combined implied better water quality than the coarse mesh alone in the Savannah River but lower water quality in the Mississippi River. These results indicate that mesh size can have a significant impact on ecological studies of macroinvertebrates in large rivers, and suggest that the finer mesh produces a more accurate estimate of the structure and density of the macroinvertebrate community. Our results also suggest caution when different mesh sizes are involved in a large river study—it may be inappropriate to contrast data produced with different mesh sizes or to combine data from different mesh sizes to create a long-term perspective.

In this paper, Newton iteration and two-level finite element algorithm are combined for solving numerically the stationary incompressible magnetohydrodynamics (MHD) under a strong uniqueness condition. The method consists of solving the nonlinear MHD system by $$m$$m Newton iterations on a coarse mesh with size $$H$$H and then computing the Stokes and Maxwell problems on a fine mesh with size $$h\ll H$$h?H. The uniform stability and optimal error estimates of both Newton iterative method and two-level Newton iterative method are given. The error analysis shows that the two-level Newton iterative solution is of the same convergence order as the Newton iterative solution on a fine grid with $$h=O(H^2)$$h=O(H2). However, the two-level Newton iterative method for solving the stationary incompressible MHD equations is simpler and more efficient than Newton iterative one. Finally, the effectiveness of the two-level Newton iterative method is illustrated by several numerical investigations.

Biochemical reaction networks are often modelled using discrete-state, continuous-time Markov chains. System statistics of these Markov chains usually cannot be calculated analytically and therefore estimates must be generated via simulation techniques. There is a well documented class of simulation techniques known as exact stochastic simulation algorithms, an example of which is Gillespie’s direct method. These algorithms often come with high computational costs, therefore approximate stochastic simulation algorithms such as the tau-leap method are used. However, in order to minimise the bias in the estimates generated using them, a relatively small value of tau is needed, rendering the computational costs comparable to Gillespie’s direct method. The multi-level Monte Carlo method (Anderson and Higham, Multiscale Model. Simul. 10:146–179, 2012) provides a reduction in computational costs whilst minimising or even eliminating the bias in the estimates of system statistics. This is achieved by first crudely approximating required statistics with many sample paths of low accuracy. Then correction terms are added until a required level of accuracy is reached. Recent literature has primarily focussed on implementing the multi-level method efficiently to estimate a single system statistic. However, it is clearly also of interest to be able to approximate entire probability distributions of species counts. We present two novel methods that combine known techniques for distribution reconstruction with the multi-level method. We demonstrate the potential of our methods using a number of examples.

Microclimate within litter bags of different mesh size: Implications for the ‘arthropod effect’ on litter decomposition