Ultra Weak Variational Formulation Research Articles

Ultra-weak variational formulation for heterogeneous Maxwell problem in the context of high performance computing

Electromagnetic simulations on large domains require a huge memory consumption. Domain decomposition methods, based on Trefftz methods, could be an answer to this issue. In this paper, we associate to heterogeneous three-dimensional Maxwell equations one variational formulation which can be obtained either by upwind fluxes or Riemann traces. We associate to this variational formulation an iterative Trefftz Krylov solver. The poor conditioning due to the use of plane wave basis functions is bypassed thanks to a compression strategy. Moreover, the developed iterative solver is accelerated thanks to a left preconditioner. The considered numerical cases illustrate the performance of this basis reduction, which leads to the consideration of an industrial case of more than 750 millions of degrees of freedom.

An Ultraweak Variational Method for Parameterized Linear Differential-Algebraic Equations

We investigate an ultraweak variational formulation for (parameterized) linear differential-algebraic equations with respect to the time variable which yields an optimally stable system. This is used within a Petrov-Galerkin method to derive a certified detailed discretization which provides an approximate solution in an ultraweak setting as well as for model reduction with respect to time in the spirit of the Reduced Basis Method. A computable sharp error bound is derived. Numerical experiments are presented that show that this method yields a significant reduction and can be combined with well-known system theoretic methods such as Balanced Truncation to reduce the size of the DAE.

Local strategies for improving the conditioning of the plane-wave Ultra-Weak Variational Formulation

Element-wise techniques based on SVD or QR Decomposition completely get rid of the usual ill-conditioning inherent to the plane-wave discretizations of the Ultra-Weak Variational Formulation (UWVF) for the Helmholtz equation. Associated preconditioning strategies lead to very low condition numbers of the corresponding linear system matrices, without any limitation on the number of plane waves per element. In addition, some of these procedures have the advantage of considerably reducing the size of the final system to be solved without altering the accuracy of the numerical solution.

The double adaptivity paradigm: (How to circumvent the discrete inf–sup conditions of Babuška and Brezzi)

We present an efficient implementation of the double adaptivity algorithm of Cohen et al. (2012) within the setting of the Petrov–Galerkin method with optimal test functions. We apply this method to the ultraweak variational formulation of a general linear variational problem discretized with the standard Galerkin finite element method. As an example, we demonstrate the feasibility of the method in the context of the convection-dominated diffusion problem. The presented ideas, however, apply to virtually any well-posed system of first-order partial differential equations, including singular perturbation problems.

Analysis of non-conforming DPG methods on polyhedral meshes using fractional Sobolev norms

The work is concerned with two problems: (a) analysis of a discontinuous Petrov–Galerkin (DPG) method set up in fractional energy spaces, (b) use of the results to investigate a non-conforming version of the DPG method for general polyhedral meshes. We use the ultraweak variational formulation for the model Laplace equation. The theoretical estimates are supported with 3D numerical experiments.

An ultraweak formulation of the Reissner–Mindlin plate bending model and DPG approximation

We develop and analyze an ultraweak variational formulation of the Reissner–Mindlin plate bending model both for the clamped and the soft simply supported cases. We prove well-posedness of the formulation, uniformly with respect to the plate thickness t. We also prove weak convergence of the Reissner–Mindlin solution to the solution of the corresponding Kirchhoff–Love model when $$t\rightarrow 0$$. Based on the ultraweak formulation, we introduce a discretization of the discontinuous Petrov–Galerkin type with optimal test functions (DPG) and prove its uniform quasi-optimal convergence. Our theory covers the case of non-convex polygonal plates. A numerical experiment for some smooth model solutions with fixed load confirms that our scheme is locking free.

On the pressure Poisson equation for the Stokes system

AbstractFor the numerical solution of the pressure Poisson equation, we consider an ultra–weak variational formulation and a related finite element method of Galerkin–Petrov type. While this allows a piecewise constant approximation of the pressure, the test functions have to be sufficiently smooth – in particular, we use modified B–splines of second order to guarantee conformity and stability, and optimal convergence. This novel formulation allows the computation of the pressure from either a given, measured, or precomputed velocity field as it appears in many applications in fluid mechanics.

An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation

We develop and analyze an ultraweak variational formulation for a variant of the Kirchhoff-Love plate bending model. Based on this formulation, we introduce a discretization of the discontinuous Petrov-Galerkin type with optimal test functions (DPG). We prove well-posedness of the ultraweak formulation and quasi-optimal convergence of the DPG scheme. The variational formulation and its analysis require tools that control traces and jumps in $H^2$ (standard Sobolev space of scalar functions) and $H(\mathrm{div\,Div})$ (symmetric tensor functions with $L_2$-components whose twice iterated divergence is in $L_2$), and their dualities. These tools are developed in two and three spatial dimensions. One specific result concerns localized traces in a dense subspace of $H(\mathrm{div\,Div})$. They are essential to construct basis functions for an approximation of $H(\mathrm{div\,Div})$. To illustrate the theory we construct basis functions of the lowest order and perform numerical experiments for a smooth and a singular model solution. They confirm the expected convergence behavior of the DPG method both for uniform and adaptively refined meshes.

Comparisons of three kinds of plane wave methods for the Helmholtz equation and time-harmonic Maxwell equations with complex wave numbers

In this paper we are concerned with some plane wave discretization methods of the Helmholtz equation and time-harmonic Maxwell equations with complex wave numbers. We design two new variants of the variational theory of complex rays and the ultra weak variational formulation for the discretization of these types of equations, respectively. The well posedness of the approximate solutions generated by the two methods is derived. Moreover, based on the PWLS–LSFE method introduced in Hu and Yuan (2018), we extend these two methods (VTCR method and UWVF method) combined with local spectral element to discretize nonhomogeneous Helmholtz equation and Maxwell’s equations. The numerical results show that the resulting approximate solution generated by the UWVF method is clearly more accurate than that generated by the VTCR method.

Superconvergence in a DPG method for an ultra-weak formulation

In this work we study a DPG method for an ultra-weak variational formulation of a reaction–diffusion problem. We improve existing a priori convergence results by sharpening an approximation result for the numerical flux. By duality arguments we show that higher convergence rates for the scalar field variable are obtained if the polynomial order of the corresponding approximation space is increased by one. Furthermore, we introduce a simple elementwise postprocessing of the solution and prove superconvergence. Numerical experiments indicate that the obtained results are valid beyond the underlying model problem.

The ultra weak variational formulation for the modified mild-slope equation

• The Ultra Weak Variational Formulation (UWVF) has been used successfully on the modified mild slope equation. • The UWVF was improved by using the main direction with the ray tracing method. • The UWVF solves some of the difficulties that arise when Finite Element Method is applied. This paper is devoted to evaluate a new alternative numerical method to capture accurately the diffraction-refraction process of waves in coastal areas. The modified mild-slope equation (MMSE) has been used to predict the water wave transformation when waves approach the shoreline. We perform numerical simulations in order to illustrate the efficiency of the ultra weak variational formulation (UWVF) method in comparison with the finite elements method (FEM). The UWVF method uses plane wave solutions on each element and has been shown to reduce the computational complexity at high wave numbers. We also present an alternative method to seek the angle of attack of the wave front on the domain boundary and show that the UWVF method reproduces effectively the numerical experimental data. We compare the FEM and the UWVF method for the MMSE and show that the UWVF method solves some of the difficulties that arise when the FEM is applied.

Performance analysis of the ultra weak variational formulation to compute electromagnetic fields on nonuniform meshes

AbstractThe purpose of this work is to apply a previously developed ultra weak variational formulation (UWVF) to the solution of the Helmholtz equation in electromagnetic scattering problems. The variational form is based on discontinuous Galerkin with plane wave basis functions in 2‐dimensional finite element mesh. We present a general strategy, which allows to adjust the number of wave directions based on both the element size and the local conditioning number. This provides a cost‐effective mechanism for application of UWVF using nonuniform meshes. Through simulation of scattering of a dielectric‐coated circular cylinder, it is shown that the process significantly improves the UWVF performance.

DPG Method with Optimal Test Functions for a Fractional Advection Diffusion Equation

We develop an ultra-weak variational formulation of a fractional advection diffusion problem in one space dimension and prove its well-posedness. Based on this formulation, we define a DPG approximation with optimal test functions and show its quasi-optimal convergence. Numerical experiments confirm expected convergence properties, for uniform and adaptively refined meshes.

A Time-Stepping DPG Scheme for the Heat Equation

Abstract We introduce and analyze a discontinuous Petrov–Galerkin method with optimal test functions for the heat equation. The scheme is based on the backward Euler time stepping and uses an ultra-weak variational formulation at each time step. We prove the stability of the method for the field variables (the original unknown and its gradient weighted by the square root of the time step) and derive a Céa-type error estimate. For low-order approximation spaces this implies certain convergence orders when time steps are not too small in comparison with mesh sizes. Some numerical experiments are reported to support our theoretical results.

A Posteriori Error Estimation of $hp$-$dG$ Finite Element Methods for Highly Indefinite Helmholtz Problems

In this paper, we will consider an $hp$-finite elements discretization of a highly indefinite Helmholtz problem by some ${dG}$-formulation which is based on the ultra-weak variational formulation by Cessenat and Deprés. We will introduce an a posteriori error estimator and derive reliability and efficiency estimates which are explicit with respect to the wavenumber and the discretization parameters $h$ and $p$. In contrast to the conventional conforming finite element method for indefinite problems, the ${dG}$-formulation is unconditionally stable and the adaptive discretization process may start from a very coarse initial mesh. Numerical experiments will illustrate the efficiency and robustness of the method.

A robust Petrov–Galerkin discretisation of convection–diffusion equations

A Petrov–Galerkin discretisation is studied of an ultra-weak variational formulation of the convection–diffusion equation in a mixed form. To arrive at an implementable method, the truly optimal test space has to be replaced by its projection onto a finite dimensional test search space. To prevent that this latter space has to be taken increasingly large for vanishing diffusion, a formulation is constructed that is well-posed in the limit case of a pure transport problem. Numerical experiments show approximations that are very close to the best approximations to the solution from the trial space, uniformly in the size of the diffusion term.

Error Estimation for Numerical Methods Using the Ultra Weak Variational Formulation in Model of Near Field Scattering Problem

In this paper, we investigate the use of ultra weak variational formulation to solve a wave scattering problem in near field optics. In order to capture the sub-scale features of waves, we utilize evanescent wave functions together with plane wave functions to approximate the local properties of the field. We analyze the global convergence and give an error estimation of the method. Numerical examples are also presented to demonstrate the effectiveness of the strategy.

Implementation of an interior point source in the ultra weak variational formulation through source extraction

The Ultra Weak Variational Formulation (UWVF) is a powerful numerical method for the approximation of acoustic, elastic and electromagnetic waves in the time-harmonic regime. The use of Trefftz-type basis functions incorporates the known wave-like behaviour of the solution in the discrete space, allowing large reductions in the required number of degrees of freedom for a given accuracy, when compared to standard finite element methods. However, the UWVF is not well disposed to the accurate approximation of singular sources in the interior of the computational domain. We propose an adjustment to the UWVF for seismic imaging applications, which we call the Source Extraction UWVF. Differing fields are solved for in subdomains around the source, and matched on the inter-domain boundaries. Numerical results are presented for a domain of constant wavenumber and for a domain of varying sound speed in a model used for seismic imaging.

Dispersion analysis of plane wave discontinuous Galerkin methods

SUMMARYThe plane wave DG (PWDG) method for the Helmholtz equation was introduced and analyzed in [GITTELSON, C., HIPTMAIR, R., AND PERUGIA, I. Plane wave discontinuous Galerkin methods: analysis of the h‐version. Math. Model. Numer. Anal. 43 (2009), 297–331] as a generalization of the so‐called ultra‐weak variational formulation, see [O. CESSENAT AND B. DESPRéS, Application of an ultra weak variational formulation of elliptic PDEs to the two‐dimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp. 255–299]. The method relies on Trefftz‐type local trial spaces spanned by plane waves of different directions, and links cells of the mesh through numerical fluxes in the spirit of DG methods.We conduct a partly empirical dispersion analysis of the method in a discrete translation‐invariant setting by studying the mismatch of wave numbers of discrete and continuous plane waves traveling in the same direction. We find agreement of the wave numbers for directions represented in the local trial spaces. For other directions, the PWDG methods turn out to incur both phase and amplitude errors. This manifests itself as a pollution effect haunting the h‐version of the method. Our dispersion analysis allows a quantitative prediction of the strength of this effect and its dependence on the wave number and number of plane waves. Copyright © 2014 John Wiley & Sons, Ltd.

A Plane-Wave Least-Squares Method for Time-Harmonic Maxwell's Equations in Absorbing Media

In this paper we discuss numerical methods for solving time-harmonic Maxwell equations in three-dimensional absorbing media. We propose a new variational problem on the space spanned by the plane-wave basis functions for the discretization of these types of equations. The variational problem is derived mathematically from a minimization problem, and the resulting stiffness matrix is Hermitian positive definite. A simple domain decomposition preconditioner is introduced for the system generated by the proposed variational formula. Numerical results indicate that the approximate solution generated by the proposed discretization method is clearly more accurate than that generated by the ultra weak variational formulation method, and the proposed preconditioner is effective.