Quasi-optimal Error Estimates Research Articles

Note on quasi-optimal error estimates for the pressure for shear-thickening fluids

In this paper, we derive quasi-optimal a priori error estimates for the kinematic pressure for a Local Discontinuous Galerkin (LDG) approximation of steady systems of $p$-Navier--Stokes type in the case of shear-thickening, \textit{i.e.}, $p>2$, imposing a new Muckenhoupt regularity condition on the viscosity of the fluid, which is mild in the two-dimensional case but potentially restrictive in the three-dimensional case.

Numerical treatment for some abstract degenerate second-order evolutionary problem

This paper addresses the numerical analysis of a class of a degenerate second-order evolution equations. We employ a finite element method for spatial discretization and a family of implicit finite difference schemes for time discretization. Introducing a stabilization parameter, denoted by θ, we propose a well-posed fully-discrete scheme. Sufficient conditions for its well-posedness and for quasi-optimal error estimates are established. The abstract theory is illustrated through the application to the degenerate wave equation, and numerical results validate our theoretical findings.

An [formula omitted] error analysis of a hybrid discontinuous mixed Galerkin method for linear viscoelasticity

We present a hybrid discontinuous Galerkin method for the velocity/stress formulation of Zener’s model in dynamic viscoelasticity. Our approach utilizes a spatial discretization that enforces strongly the symmetry of the stress tensor, and that allows for efficient handling of heterogeneous materials comprising both purely elastic and viscoelastic components. We provide an hp error analysis of the semidiscrete scheme, which yields quasi-optimal error estimates for the stress tensor and sub-optimal error estimates for the velocity field in the L2-norm. Next, we apply the Crank–Nicolson rule as a time-stepping scheme and analyze its primary convergence properties. Finally, we present the results of numerical experiments to validate our approach and confirm the theoretical rates of convergence.

Numerical analysis of nonlinear degenerate parabolic problems with application to eddy current models

This paper deals with the numerical analysis for a family of nonlinear degenerate parabolic problems. The model is spatially discretized using a finite element method; an implicit Euler scheme is employed for time discretization. We deduce sufficient conditions to ensure that the fully discrete problem has a unique solution and to prove quasi-optimal error estimates for the approximation. Finally, we propose a nonlinear degenerate parabolic problem that arises from electromagnetic applications in conductive nonlinear magnetic media and deduce its solubility and convergence by using the developed abstract theory, including some numerical results to confirm the obtained theoretical results.

Numerical analysis of a FEM based on a time-primitive of the electric field for solving a nonlinear transient eddy current problem

The aim of this paper is to analyze from a mathematical and a numerical point of view a formulation of a transient eddy current problem in terms of a time-primitive of the electric field in a bounded domain with ferromagnetic conductors. To this aim, we introduce a Lagrange multiplier to impose the free-divergence condition in the isolated domain. Thus, we obtain a nonlinear degenerate parabolic problem in mixed form and prove its well-posedness. Then, we propose a fully-discrete scheme by using an implicit Euler scheme for time-discretization and a finite element method based on edge and nodal elements for the spatial discretization. We prove quasi-optimal error estimates for the approximation and include some numerical examples to confirm the obtained theoretical results.

Error estimates for total-variation regularized minimization problems with singular dual solutions

Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems using the Crouzeix–Raviart element require the existence of a Lipschitz continuous dual solution, which is not generally given. We provide analytic proofs showing that the Lipschitz continuity of a dual solution is not necessary, in general. Using the Lipschitz truncation technique, we, in addition, derive error estimates that depend directly on the Sobolev regularity of a given dual solution.

Singular solutions, graded meshes,and adaptivity for total-variation regularized minimization problems

Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems require the existence of a Lipschitz continuous dual solution. We discuss the validity of this condition and devise numerical methods using locally refined meshes that lead to improved convergence rates despite the occurrence of discontinuities. It turns out that linear convergence is possible on suitably constructed meshes.

FULLY-DISCRETE FINITE ELEMENT APPROXIMATION FOR A FAMILY OF DEGENERATE PARABOLIC PROBLEMS

The aim of this work is to show an abstract framework to analyze the numerical approximation by using a finite element method in space and a BackwardEuler scheme in time of a family of degenerate parabolic problems. We deduce sufficient conditions to ensure that the fully-discrete problem has a unique solution and to prove quasi-optimal error estimates for the approximation. Finally, we show a degenerate parabolic problem which arises from electromagnetic applications and deduce its well-posedness and convergence by using the developed abstract theory, including numerical tests to illustrate the performance of the method and confirm the theoretical results.

Quasi-optimal Nonconforming Approximation of Elliptic PDEs with Contrasted Coefficients and $$H^{1+{r}}$$, $${r}>0$$, Regularity

In this paper, we investigate the approximation of a diffusion model problem with contrasted diffusivity for various nonconforming approximation methods. The essential difficulty is that the Sobolev smoothness index of the exact solution may be just barely larger than 1. The lack of smoothness is handled by giving a weak meaning to the normal derivative of the exact solution at the mesh faces. We derive robust and quasi-optimal error estimates. Quasi-optimality means that the approximation error is bounded, up to a generic constant, by the best approximation error in the discrete trial space, and robustness means that the generic constant is independent of the diffusivity contrast. The error estimates use a mesh-dependent norm that is equivalent, at the discrete level, to the energy norm and that remains bounded as long as the exact solution has a Sobolev index strictly larger than 1. Finally, we briefly show how the analysis can be extended to the Maxwell’s equations.

Equilibrium analysis of an immersed rigid leaflet by the virtual element method

We study, both theoretically and numerically, the equilibrium of a hinged rigid leaflet with an attached rotational spring, immersed in a stationary incompressible fluid within a rigid channel. Through a careful investigation of the properties of the domain functional describing the angular momentum exerted by the fluid on the leaflet (which depends on both the leaflet angular position and its thickness), we identify sufficient conditions on the spring stiffness function for the existence (and uniqueness) of equilibrium positions. This study resorts to techniques from shape differential calculus. We propose a numerical technique that exploits the mesh flexibility of the Virtual Element Method (VEM). A (polygonal) computational mesh is generated by cutting a fixed background grid with the leaflet geometry, and the problem is then solved with stable VEM Stokes elements of degrees [Formula: see text] and [Formula: see text] combined with a bisection algorithm. We prove quasi-optimal error estimates and present a large array of numerical experiments to document the accuracy and robustness with respect to degenerate geometry of the proposed methodology.

Fully discrete finite element approximation for a family of degenerate parabolic mixed equations

The aim of this work is to show an abstract framework to analyze the numerical approximation for a family of linear degenerate parabolic mixed equations by using a finite element method in space and a Backward-Euler scheme in time. We consider sufficient conditions to prove that the fully-discrete problem has a unique solution and prove quasi-optimal error estimates for the approximation. Furthermore, we show that mixed finite element formulations arising from dynamics fluids (time-dependent Stokes problem) and from electromagnetic applications (eddy current models), can be analyzed as applications of the developed theory. Finally, we include numerical tests to illustrate the performance of the method and confirm the theoretical results.

A boundary and finite element coupling for a magnetically nonlinear eddy current problem

The aim of this paper is to provide a mathematical and numerical analysis for a FEM-BEM coupling approximation of a magnetically nonlinear eddy current formulation by using FEM only on the conducting domain, and by imposing the integral conditions on its boundary. The nonlinear relationship between flux density and the magnetic field intensity is given by a physical parameter called magnetic reluctivity, which is assumed to depend on the Euclidean norm of the magnetic induction in the conducting domain. We use the nonlinear monotone operator theory for parabolic equations to show that the continuous formulation obtained for the coupling is a well-posed problem. Furthermore, we use Nédélec edge elements, standard nodal finite elements, and a backward-Euler time scheme, to obtain a fully discrete formulation and to prove quasi-optimal error estimates.

Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws

In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh2 and the second order TVD-RK scheme needs $ \tau \le \rho {h}^{\frac{4}{3}}$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h.

Frequency-explicit asymptotic error estimates for a stress-pressure formulation of a time harmonic fluid-solid interaction problem

This paper deals with a mixed finite element method to solve the interior solid–fluid interaction problem in harmonic regime. The main variables of our formulation are the stress tensor in the solid and the pressure in the fluid domain. The problem is shown to be well-posed and the continuous functional calculus theorem is used to obtain wavenumber-explicit stability estimates. We discretize the problem by using the mixed finite element method of Arnold–Falk–Winther in the solid and the classical Lagrange finite element in the fluid. We obtain quasi-optimal error estimates under a suitable restriction on the mesh size. Finally, our analysis is illustrated with some numerical experiments.

A discontinuous Galerkin method for a time-harmonic eddy current problem

We introduce and analyze a discontinuous Galerkin method for a time-harmonic eddy current problem formulated in terms of the magnetic field. The scheme is obtained by putting together a DG method for the approximation of the vector field variable representing the magnetic field in the conductor and a DG method for the Laplace equation whose solution is a scalar magnetic potential in the insulator. The transmission conditions linking the two problems are taken into account weakly in the global discontinuous Galerkin scheme. We prove that the numerical method is uniformly stable and obtain quasi-optimal error estimates in the DG-energy norm.

Error analysis of a FVEM for fractional order evolution equations with nonsmooth initial data

In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative of order α ∈ (0, 1) in time. Improving upon earlier results [Karaa et al., IMA J. Numer. Anal. 37 (2017) 945–964], error estimates in L2 (Ω)- and H1 (Ω)-norms for the semidiscrete problem with smooth and mildly smooth initial data, i.e., v ∈ H2(Ω) ∩ H01(Ω) and v ∈ H01(Ω) are established. For nonsmooth data, that is, v ∈ L2 (Ω), the optimal L2 (Ω)-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Superconvergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the L∞ (Ω)-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.

A penalty method for a linear Koiter shell model

In this paper a penalized method and its approximation by finite element method are proposed to solve Koiter’s equations for a thin linearly elastic shell. In addition to existence and uniqueness results of solutions of the continuous and the discrete problems we derive some a priori error estimates. We are especially interested in the behavior of the solution when the penalty parameter goes to zero. We propose here a new formulation that leads to a quasi optimal and uniform error estimate with respect to the penalized parameter. In other words, we are able to show that this method converges uniformly with respect to the penalized parameter and to the mesh size. Numerical tests that validate and illustrate our approach are given.

A nonlinear consistent penalty method weakly enforcing positivity in the finite element approximation of the transport equation

We devise and analyze a new stabilized finite element method to solve the first-order transport (or advection–reaction) equation. The method combines the usual Galerkin/Least-Squares approach to achieve stability with a nonlinear consistent penalty term inspired by recent discretizations of contact problems to weakly enforce a positivity condition on the discrete solution. We prove the existence and the uniqueness of the discrete solution. Then we establish quasi-optimal error estimates for smooth solutions bounding the usual error terms in the Galerkin/Least-Squares error analysis together with the violation of the maximum principle by the discrete solution. Numerical examples are presented to illustrate the performances of the method.

A Robust DPG Method for Singularly Perturbed Reaction-Diffusion Problems

We present and analyze a discontinuous Petrov-Galerkin method with optimal test functions for a reaction-dominated diffusion problem in two and three space dimensions. We start with an ultra-weak formulation that comprises parameters $\alpha$, $\beta$ to allow for general $\varepsilon$-dependent weightings of three field variables ($\varepsilon$ being the small diffusion parameter). Specific values of $\alpha$ and $\beta$ imply robustness of the method, that is, a quasi-optimal error estimate with a constant that is independent of $\varepsilon$. Moreover, these values lead to a norm for the field variables that is known to be balanced in $\varepsilon$ for model problems with typical boundary layers. Several numerical examples underline our theoretical estimates and reveal stability of approximations even for very small $\varepsilon$.

Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem

We aim to provide a finite element analysis for the elastoacoustic vibration problem. We use a dual-mixed variational formulation for the elasticity system and combine the lowest order Lagrange finite element in the fluid domain with the reduced symmetry element known as PEERS and introduced for linear elasticity in [1]. We show that the resulting global nonconforming scheme provides a correct spectral approximation and we prove quasi-optimal error estimates. Finally, we confirm the asymptotic rates of convergence by numerical experiments.