Efficient discretisation and domain decomposition preconditioners for incompressible fluid mechanics

Solving the linear elasticity and Stokes equations by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the non standard interface conditions are naturally defined at the boundary between elements. In this manuscript we present the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. An analysis of the boundary value problem with non standard condition is provided as well as the numerical evidence showing the advantages of the new method. Furthermore, we present and analyse a stabilisation method for the presented discretisation that allows the use of the same polynomial degrees for velocity and pressure discrete spaces. The original definition of the domain decomposition preconditioners is one-level, this is, the preconditioner is built only using the solution of local problems. This has the undesired consequence that the results are not scalable, it means that the number of iterations needed to reach convergence increases with the number of subdomains. This is the reason why we have also introduced, and tested numerically, two-level preconditioners. Such preconditioners use a coarse space in their construction. We consider two finite element discretisations, namely, the hybrid discontinuous Galerkin and Taylor-Hood discretisations for the nearly incompressible elasticity problems and Stokes equations.Solving the linear elasticity and Stokes equations by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the non standard interface conditions are naturally defined at the boundary between elements. In this manuscript we present the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. An analysis of the boundary value problem with non standard condition is provided as well as the numerical evidence showing the advantages of the new method. Furthermore, we present and analyse a stabilisation method for the presented discretisation that allows the use of the same polynomial degrees for velocity and pressure discrete spaces. The original definition of the domain decomposition preconditioners is one-level, this is, the preconditioner is built only using the solution of local problems. This has the undesired consequence that the results are not scalable, it means that the number of iterations needed to reach convergence increases with the number of subdomains. This is the reason why we have also introduced, and tested numerically, two-level preconditioners. Such preconditioners use a coarse space in their construction. We consider two finite element discretisations, namely, the hybrid discontinuous Galerkin and Taylor-Hood discretisations for the nearly incompressible elasticity problems and Stokes equations

Stabilised hybrid discontinuous Galerkin methods for the Stokes problem with non-standard boundary conditions

In several studies it has been observed that, when using stabilised $\mathbb{P}_k^{}\times\mathbb{P}_k^{}$ elements for both velocity and pressure, the error for the pressure is smaller, or even of a higher order in some cases, than the one obtained when using inf-sup stable $\mathbb{P}_k^{}\times\mathbb{P}_{k-1}^{}$ (although no formal proof of either of these facts has been given). This increase in polynomial order requires the introduction of stabilising terms, since the finite element pairs used do not stability the inf-sup condition. With this motivation, we apply the stabilisation approach to the hybrid discontinuous Galerkin discretisation for the Stokes problem with non-standard boundary conditions

Maximum likelihood estimation for discrete exponential families and random graphs

We characterize the existence of the maximum likelihood estimator for discrete exponential families. Our criterion is simple to apply as we show in various settings, most notably for exponential models of random graphs. As an application, we point out the size of independent identically distributed samples for which the maximum likelihood estimator exists with high probability.Comment: 21 pages, minor editorial changes, added connections to the criterion of Barndorff-Nielsen and linear programmin

Stabilised hybrid discontinuous Galerkin methods for the Stokes problem with non-standard boundary conditions

In several studies it has been observed that, when using stabilised ℙk×ℙk elements for both velocity and pressure, the error for the pressure is smaller, or even of a higher order in some cases, than the one obtained when using inf-sup stable ℙk×ℙk−1 (although no formal proof of either of these facts has been given). This increase in polynomial order requires the introduction of stabilising terms, since the finite element pairs used do not guarantee the inf-sup condition. With this motivation, we apply the stabilisation approach to the hybrid discontinuous Galerkin discretisation for the Stokes problem with non-standard boundary conditions

Numerical assessment of two-level domain decomposition preconditioners for incompressible Stokes and elasticity equations. ETNA - Electronic Transactions on Numerical Analysis

Solving the linear elasticity and Stokes equations by an optimal domain decomposition method derived algebraically involves the use of non-standard interface conditions. The one-level domain decomposition preconditioners are based on the solution of local problems. This has the undesired consequence that the results are not scalable, which means that the number of iterations needed to reach convergence increases with the number of subdomains. This is the reason why in this work we introduce, and test numerically, two-level preconditioners. Such preconditioners use a coarse space in their construction. We consider the nearly incompressible elasticity problems and Stokes equations, and discretise them by using two finite element methods, namely, the hybrid discontinuous Galerkin and Taylor-Hood discretisations

A domain decomposition method for isogeometric multi-patch problems with inexact local solvers

In Isogeometric Analysis, the computational domain is often described as multi-patch, where each patch is given by a tensor product spline/NURBS parametrization. In this work we propose a FETI-like solver where local inexact solvers exploit the tensor product structure at the patch level. To this purpose, we extend to the isogeometric framework the so-called All-Floating variant of FETI, that allows us to use the Fast Diagonalization method at the patch level. We construct then a preconditioner for the whole system and prove its robustness with respect to the local mesh-size $h$ and patch-size $H$ (i.e., we have scalability). Our numerical tests confirm the theory and also show a favourable dependence of the computational cost of the method from the spline degree $p$.Comment: 19 pages, 1 figur

Hybrid discontinuous Galerkin discretisation and domain decomposition preconditioners for the Stokes problem

Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of nonstandard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the nonstandard interface conditions are naturally defined at the boundary between elements. In this paper, we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with nonstandard boundary conditions. The full stability and convergence analysis of the discretisation method is presented, and the results are corroborated by numerical experiments. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments

Coupling finite and boundary element methods to solve the Poisson–Boltzmann equation for electrostatics in molecular solvation

The Poisson–Boltzmann equation is widely used to model electrostatics in molecular systems. Available software packages solve it using finite difference, finite element, and boundary element methods, where the latter is attractive due to the accurate representation of the molecular surface and partial charges, and exact enforcement of the boundary conditions at infinity. However, the boundary element method is limited to linear equations and piecewise constant variations of the material properties. In this work, we present a scheme that couples finite and boundary elements for the linearised Poisson–Boltzmann equation, where the finite element method is applied in a confined solute region and the boundary element method in the external solvent region. As a proof‐of‐concept exercise, we use the simplest methods available: Johnson–Nédélec coupling with mass matrix and diagonal preconditioning, implemented using the Bempp‐cl and FEniCSx libraries via their Python interfaces. We showcase our implementation by computing the polar component of the solvation free energy of a set of molecules using a constant and a Gaussian‐varying permittivity. As validation, we compare against well‐established finite difference solvers for an extensive binding energy data set, and with the finite difference code APBS (to 0.5%) for Gaussian permittivities. We also show scaling results from protein G B1 (955 atoms) up to immunoglobulin G (20,148 atoms). For small problems, the coupled method was efficient, outperforming a purely boundary integral approach. For Gaussian‐varying permittivities, which are beyond the applicability of boundary elements alone, we were able to run medium to large‐sized problems on a single workstation. The development of better preconditioning techniques and the use of distributed memory parallelism for larger systems remains an area for future work. We hope this work will serve as inspiration for future developments that consider space‐varying field parameters, and mixed linear‐nonlinear schemes for molecular electrostatics with implicit solvent models