October 10 – Peter Minev: Recent splitting schemes for the incompressible Navier-Stokes equations

Peter Minev: Tuesday 10 October at 11 am, A415 Inria Paris.

The presentation will be focused on two classes of recently developed splitting schemes for the Navier-Stokes equations.
The first class is based on the classical artificial compressibility (AC) method. The original method proposed by J. Shen in 1995 reduces the solution of the incompressible Navier-Stokes equations to a set of two or three parabolic problems in 2D and 3D correspondingly. Unfortunately, its accuracy is limited to first order in time and can be extended further only if the resulting scheme involves an elliptic problem for the velocity vector. Recently, together with J.L. Guermond (Texas A&M University) we proposed a scheme that extends the AC method to any order in time using a bootstrapping approach to the incompressibility constraint that essentially requires to solve only a set of parabolic equations for the velocity. The conditioning of the corresponding linear systems is therefore O(Δth^-2). This is generally better than solving a parabolic equation for the velocity and an elliptic equation for the pressure required by the various projection schemes that are perhaps the most popular approach at present. Besides, the bootstrapping algorithm allows to achieve any order in time, subject to some initialization conditions, in contrast to the projection methods whose accuracy seems to be essentially limited to second order on the velocity in the L2 norm.
The second class of methods is based on a novel approach to the Navier-Stokes equations that reformulates them in terms of stress variables. It was developed in a recent paper together with P. Vabishchevich (Russian Academy of Sciences). The main advantage of such an approach becomes clear when it is applied to fluid-structure interaction problems since in such case the problems for the fluid and the structure, both written in terms of …

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September 18 – Théophile Chaumont: High order finite element methods for the Helmholtz equation in highly heterogeneous media

Théophile Chaumont: Monday 18 September at 1 pm, A415 Inria Paris.

Time-harmonic wave propagation problems are costly to solve numerically
since the corresponding PDE operators are not strongly elliptic, and as a result,
discretization methods might become unstable. Specifically, the finite element solution is
quasi-optimal (almost as good as the best approximation the finite element space
can provide) only under restrictive assumptions on the mesh size. If the mesh
size is too large, stability is lost, and the finite element solution can become
completely inaccurate, even when the best approximation is. This phenomenon is called the
“pollution effect” and becomes more important for larger frequencies.

For the case of wave propagation problems in homogeneous media, it is known that
high order finite element methods are less sensitive to the pollution effect. For this
reason, they are employed in a wide range of applications, as the corresponding linear
systems are smaller and easier to solve.

In this talk, we investigate the use of high order finite element methods to solve
wave propagation problems in highly heterogeneous media. Since the heterogeneities
of the medium can exhibit small scale features, we consider “non-fitting” meshes,
that are not aligned with the physical interfaces of the medium. Instead, the parameters
defining the medium of propagation can be discontinuous inside each element. We propose
a convergence analysis and draw two main conclusions:
– the asymptotic convergence rate of the proposed finite element method is suboptimal
due to the lack of regularity of the solution inside each cell
– the pollution effect is greatly reduced by increasing the order of discretization.
We illustrate our main conclusions with geophysical application benchmarks.
These examples confirm that higher order methods …

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June 22 – Valentine Rey: Goal-oriented error control within non-overlapping domain decomposition methods to solve elliptic problems

Valentine Rey: Thursday 22 June at 3 pm, A415 Inria Paris.
Domain decomposition methods are robust and efficient methods to solve mechanical problems with several million degrees of freedom. Taking advantage of increasing performances of computers, they exploit the clusters-parallel architecture and are numerically scalable. Verification has been widely developed since 1980’s and proposes tools to estimate the distance between the unknown exact solution of continuous problem and the computed solution. Among those techniques, estimators based on error in constitutive relation provide constant-free upper bounds and are available for varied range of problems.
In this talk, we present techniques for steering parallel computation by objective of accuracy on quantities of interest. It relies on a parallel error estimator that provides strict guaranteed upper bound and separates the algebraic error (due to the use of iterative solver) from the discretization error (due to the finite element method). This estimator enables to adapt the solver’s stopping criterion to the discretization, which avoids over resolution and useless iterations. In [*], the estimator is used for goal-oriented error estimation and classical bounds for quantities of interest are rewritten in order to separate sources of error. Finally, we benefit the information provided by the error estimator and the Krylov subspaces built during the resolution to set an auto-adaptive strategy (adaptive remeshing and recycling search directions).

*V. Rey, P. Gosselet, C. Rey, Strict bounding of quantities of interest in computations based on domain decomposition, Computer Methods in Applied Mechanics and Engineering. 2015 Apr 15;287:212-28

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29 June – Gouranga Mallik: A priori and a posteriori error control for the von Karman equations

Gouranga Mallik: Thursday 29 June at 3pm, A415 Inria Paris.
In this work we consider a priori and a posteriori error control for the nonsingular solution of von Karman plate bending problem. Conforming and nonconforming finite element methods are employed. Existence, uniqueness and error estimates for the discrete solution are presented. We discuss an abstract framework for a posteriori error control which includes conforming and nonconforming finite element methods. This allows us to compute reliable and efficient local estimators. The key ingredients in establishing well-posedness of the discrete problem rely on the linearization of the continuous problem and suitable enrichment operator. Numerical experiments are performed to justify the theoretical results.

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15 June – Patrik Daniel: An adaptive hp-refinement strategy with computable guaranteed error reduction factors

Patrik Daniel: Thursday 15 June at 3:30pm, A415 Inria Paris.
We propose a new practical adaptive refinement strategy for hp-finite element approximations of elliptic problems. Following some recent theoretical developments in polynomial-degree-robust a posteriori error analysis, we solve two complementary classes of discrete local problems on the vertex-based patches. The first class involves the solution on each patch of a mixed finite element problem with homogeneous Neumann boundary conditions, which leads to an H(div,Ω)-conforming equilibrated flux. This in turns yields a guaranteed upper bound on the error and serves to mark elements for refinement via a Dörfler bulk criterion. The second class of local problems involves the solution, on each marked patch only, of two separate primal finite element problems with homogeneous Dirichlet boundary conditions, which serve to decide between h-, p-, or hp-refinement. Altogether, we show that these ingredients lead to a computable error reduction factor; we guarantee that while performing the hp-adaptive refinement as suggested, the error will be reduced at least by this factor on the next hp-mesh. In a series of numerical experiments in two space dimensions, we first study the accuracy of our predicted reduction factor: in particular, we measure the ratio of the predicted reduction factor relative to the true error reduction, and we find that it is very close to the optimal value of one for both smooth and singular exact solutions. Finally, we study the overall performance of the proposed hp-refinement strategy on some test cases, for which we observe effectivity indices very close to one and exponential convergence rates.

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15 June – Jad Dabaghi: Adaptive inexact semi-smooth Newton methods for a contact between two membranes

Jad Dabaghi: Thursday 15 June at 3pm, A415 Inria Paris.
We propose an adaptive inexact version of a class of semi-smooth Newton methods. As a model problem, we consider the system of variational inequalities describing the contact between two membranes and its finite element discretization. Any iterative linearization algorithm like the Newton-min, Newton-Fisher Burmeister is taken into account, as well as any iterative linear algebraic solver. We prove an a posteriori error estimate between the exact solution and the approximate solution which is valid on any step of the linearization and algebraic resolution. This estimate is based on discretization and algebraic flux reconstructions, where the latter one is obtained on a hierarchy of nested meshes. The estimate distinguishes the discretization, linearization, and algebraic components of the error and allows us to formulate adaptive stopping criteria for both solvers. Numerical experiments for the semi-smooth Newton-min algorithm in combination with the GMRES solver confirm the efficiency of the method.

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June 1 – Martin Eigel: Adaptive stochastic FE for explicit Bayesian inversion with hierarchical tensor representations

Martin Eigel: Thursday 1 June at 10:45 am, A415 Inria Paris.
We consider a class of linear PDEs with stochastic coefficients which depend on a countable (inifinite) number of random parameters. As an alternative to classical Monte Carlo sampling techniques, a functional discretisation of the stochastic space in generalised polynomial chaos may lead to significantly improved (optimal) convergence rates. However, when employed in the context of Galerkin methods, the arising algebraic systems are very high-dimensional and quickly become intractable to computations. As a matter of fact, this is an exemplary example for the curse of dimensionality with exponential growth of complexity which makes model reduction techniques inevitable. In the first part, we discuss two approaches for this: (1) a posteriori adaptivity and exploitation of sparsity of the solution, and (2) low-rank compression in a hierarchical tensor format.
In the second part, the low-rank discretisation is used as an efficient representation of the stochastic model for Bayesian inversion. This is an important application in Uncertainty Quantification where one is interested in determining the (statistics of) parameters of the model based on a set of noisy measurements. In contrast to popular sampling techniques such as MCMC, we derive an explicit representation of the posterior densities. The examined sampling-free Bayesian inversion is adaptive in all discretisation parameters. Moreover, convergence of the method is shown.

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June 1 – Joscha Gedicke: An adaptive finite element method for two-dimensional Maxwell’s equations

Joscha Gedicke: Thursday 1 June at 10 am, A415 Inria Paris.
We extend the Hodge decomposition approach for the cavity problem of two-dimensional time harmonic Maxwell’s equations to include the impedance boundary condition, with anisotropic electric permittivity and sign changing magnetic permeability. We derive error estimates for a P_1 finite element method based on the Hodge decomposition approach and develop a residual type a posteriori error estimator. We show that adaptive mesh refinement leads empirically to smaller errors than uniform mesh refinement for numerical experiments that involve metamaterials and electromagnetic cloaking. The well-posedness of the cavity problem when both electric permittivity and magnetic permeability can change sign is also discussed and verified for the numerical approximation of a flat lens experiment.

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June 1st 2017 – Quang Duc Bui – Coupled Parareal-Schwarz Waveform relaxation method for advection reaction diffusion equation in one dimension

Quang Duc Bui: Thursday 1 June at 11:30am, A415 Inria Paris.
Parareal method is a numerical method to solve time-evolution problems in parallel, which uses two propagators: the coarse – fast and inaccurate – and the fine – slow but more accurate. Instead of running the fine propagator on the whole time interval, we divide the time space into small time intervals, where we can run the fine propagator in parallel to obtain the desired solution, with the help of the coarse propagator and through parareal steps. Furthermore, each local subproblem can be solved by an iterative method, and instead of doing this local iterative method until convergence, one may perform only a few iterations of it, during parareal iterations. Propagators then become much cheaper but sharply lose their accuracy, and we hope that the convergence will be achieved across parareal iterations.

Here, we propose to couple Parareal with a well-known iterative method – Schwarz Waveform Relaxation (SWR)- with only few SWR iterations in the fine propagator and with a simple coarse propagator deduced from Backward Euler method. We present the analysis of this coupled method for 1-dimensional advection reaction diffusion equation, for this case the convergence is at least linear. We also give some numerical illustrations for 1D and 2D parabolic equations, which shows that the convergence is much faster in practice.

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May 16 – Quanling Deng: Dispersion Optimized Quadratures for Isogeometric Analysis

Quanling Deng: Tuesday 16 May at 3 pm, A415 Inria Paris.
The isogeometric analysis (IgA) is a powerful numerical tool that unifies the finite element analysis (FEA) and computer-aided design (CAD). Under the framework of FEA, IgA uses as basis functions those employed in CAD, which are capable of exactly represent various complex geometries. These basis functions are called the B-Splines or more generally the Non-Uniform Rational B-Splines (NURBS) and they lead to an approximation which may have global continuity of order up to $p-1$, where $p$ is the order of the underlying polynomial, which in return delivers more robustness and higher accuracy than that of finite elements.

We apply IgA to wave propagation and structural vibration problems to study their dispersion and spectrum properties. The dispersion and spectrum analysis are unified in the form of a Taylor expansion for eigenvalue errors. By blending optimally two standard Gaussian quadrature schemes for the integrals corresponding to the stiffness and mass, the dispersion error of IgA is minimized. The blending schemes yield two extra orders of convergence (superconvergence) in the eigenvalue errors, while the eigenfunction errors are of optimal convergence order. To analyze the eigenvalue and eigenfunction errors, the Pythagorean eigenvalue theorem (Strang and Fix, 1973) is generalized to establish an equality among the eigenvalue, eigenfunction (in L2 and energy norms), and quadrature errors.

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