Discontinuous Polynomials Research Articles

Discontinuous polynomial approximation in electrical impedance tomography with total variational regularization

In this paper, we present an alternative discrete total variation type functional for image reconstruction in electrical impedance tomography. The modified functional deals with unknown inclusions and values of conductivity. The convergence of the proposed finite element method with uniform refinement is also established: the sequence of discrete solutions contains a subsequence that converges to a solution of the continuous functional. Additionally, we propose an adaptive reconstruction algorithm in which the estimators are based on the discrete state variable, the discrete adjoint variable and the gradient of the discrete functional. A series of numerical examples are provided to illustrate the convergence behavior in diverse scenarios.

An efficient weak Galerkin FEM for third-order singularly perturbed convection-diffusion differential equations on layer-adapted meshes

In this article, we study the weak Galerkin finite element method to solve a class of a third order singularly perturbed convection-diffusion differential equations. Using some knowledge on the exact solution, we prove a robust uniform convergence of order O(N−(k−1/2)) on the layer-adapted meshes including Bakhvalov-Shishkin type, and Bakhvalov-type and almost optimal uniform error estimates of order O((N−1ln⁡N)(k−1/2)) on Shishkin-type mesh with respect to the perturbation parameter in the energy norm using high-order piecewise discontinuous polynomials of degree k. Here N is the number mesh intervals. We conduct numerical examples to support our theoretical results.

A modified weak Galerkin finite element method for the Maxwell equations on polyhedral meshes

We introduce a new numerical method for solving time-harmonic Maxwell’s equations via the modified weak Galerkin technique. The inter-element functions of the weak Galerkin finite elements are replaced by the average of the two discontinuous polynomial functions on the two sides of the polygon, in the modified weak Galerkin (MWG) finite element method. With the dependent inter-element functions, the weak curl and the weak gradient are defined directly on totally discontinuous polynomials. Optimal-order convergence of the method is proved. Numerical examples confirm the theory and show effectiveness of the modified weak Galerkin method over the existing methods.

A fragile points method with a numerical-flux-based interface debonding model to simulate the delamination migration in composite laminates

Predicting damage in composite laminates is essential for a better utilization of the material, but it is challenging especially when the inter- and intra-ply damages are coupled. In this paper, a Fragile Points Method (FPM) with a numerical-flux-based interface debonding model is proposed to simulate the laminates’ damage. The FPM is a novel meshless method using the discontinuous Galerkin weak formulations and the point-based discontinuous polynomial trial functions. In this work, an orthotropic interface debonding model has been developed based on the numerical fluxes to simulate the mixed-mode inter- and intra-ply damage. For any interior interface in an FPM model, the interface damage is triggered when the damage criterion is satisfied and the interface is separated when the fracture energy is reached. Due to the discontinuous characteristics of the FPM, the inter- and intra-ply damage modes are introduced explicitly, and thus the coupling between them can be naturally captured. In this paper, the proposed FPM is verified by four benchmark examples for Mode I, Mode II and Mixed Mode delamination. Then the proposed method is applied to simulate the delamination migration, used as an example for the inter- and intra-ply damage. The FPM results show that this method is able to provide reliable prediction on the load–displacement curves as well as the damage and crack patterns, and the mechanism governing the delamination migration is also discussed based on the FPM results.

A parameter-free mixed formulation for the Stokes equations and linear elasticity with strongly symmetric stress

In this paper, we design and analyze a parameter-free mixed method of arbitrary polynomial orders for the Stokes equations and linear elasticity problem, where the symmetry of stress is strongly imposed. Equal-order polynomials are exploited for the stress and velocity spaces in which the stress is approximated by discontinuous polynomials. In contrast, the velocity is approximated by an H(div;Ω) conforming space. As a consequence, the proposed scheme yields divergence-free velocity approximations and is pressure-robust for the Stokes equations. The stable Stokes pair are integrated to facilitate the proof of the inf-sup condition. The comprehensive convergence error estimates for all the variables are explored for the Stokes equations and linear elasticity problem. In particular, we explicitly track the dependence of the error estimates on the physical parameters and illustrate the locking-free property of the proposed scheme, which is non-trivial under the proposed formulation. The judicious balancing of the mixed formulation and the equivalent primal formulation enable us to achieve the robust error estimates for the linear elasticity problem. Several numerical experiments for the Stokes equations and linear elasticity problem are carried out to verify the proposed theories.

Error analysis of a weak Galerkin finite element method for singularly perturbed differential-difference equations

A weak Galerkin finite element method is applied to singularly perturbed delay reaction-diffusion problems. A robust uniform convergence has been proved both in the energy and balanced norms using higher-order piecewise discontinuous polynomials on Shishkin meshes. The error analysis for singularly perturbed reaction-diffusion problems with negative or positive shift in the balanced norm has appeared for the first time. The proposed method uses piecewise polynomials of order k ≥ 1 on interior of each element and piecewise constant polynomials on the end points of each element. By the Schur complement technique, the interior degrees of foredoom (DOF) can be eliminated from the discrete system resulting from the numerical scheme, and thus the degrees of freedom of the proposed method comparable with the classical finite element methods, and it is remarkably less than that of the discontinuous Galerkin method. Finally, we give various numerical experiments to verify the theoretical findings.

Four-order superconvergent CDG finite elements for the biharmonic equation on triangular meshes

In a conforming discontinuous Galerkin (CDG) finite element method, discontinuous Pk polynomials are employed. To connect discontinuous functions, the inter-element traces, {uh} and {∇uh}, are usually defined as some averages in discontinuous Galerkin finite element methods. But in this CDG finite element method, they are defined as projections of a lifted Pk+4 polynomial from four Pk polynomials on neighboring triangles. With properly chosen weak Hessian spaces, when tested by discontinuous polynomials, the variation form can have no inter-element integral, neither any stabilizer. That is, the bilinear form is the same as that of conforming finite elements for solving the biharmonic equation. Such a conforming discontinuous Galerkin finite element method converges four orders above the optimal order, i.e., the Pk solution has an O(hk+5) convergence in L2-norm, and an O(hk+3) convergence in H2-norm. A local post-process is defined, which lifts the Pk solution to a Pk+4 quasi-optimal solution. Numerical tests are provided, confirming the theory.

Convergence and optimality of an adaptive modified weak Galerkin finite element method

AbstractAn adaptive modified weak Galerkin method (AmWG) for an elliptic problem is studied in this article, in addition to its convergence and optimality. The modified weak Galerkin bilinear form is simplified without the need of the skeletal variable, and the approximation space is chosen as the discontinuous polynomial space as in the discontinuous Galerkin method. Upon a reliable residual‐based a posteriori error estimator, an adaptive algorithm is proposed together with its convergence and quasi‐optimality proved for the lowest order case. The primary tool is to bridge the connection between the modified weak Galerkin method and the Crouzeix–Raviart nonconforming finite element. Unlike the traditional convergence analysis for methods with a discontinuous polynomial approximation space, the convergence of AmWG is penalty parameter free. Numerical results are presented to support the theoretical results.

An explicit total Lagrangian Fragile Points Method for finite deformation of hyperelastic materials

This research explored a novel explicit total Lagrangian Fragile Points Method (FPM) for finite deformation of hyperelastic materials. In contrast to mesh-based methods, where mesh distortion may pose numerical challenges, meshless methods are more suitable for large deformation modelling since they use enriched shape functions for the approximation of displacements. However, this comes at the expense of extra computational overhead and higher-order quadrature is required to obtain accurate results. In this work, the novel meshless method FPM was used to derive an explicit total Lagrangian algorithm for finite deformation. FPM uses simple one-point integration for exact integration of the Galerkin weak form since it employs simple discontinuous polynomials as trial and test functions, leading to accurate results even with single-point quadrature. The proposed method was evaluated by comparing it with FEM in several case studies considering both the extension and compression of a hyperelastic material. It was demonstrated that FPM maintained good accuracy even for large deformations where FEM failed to converge.

Error estimations of a weak Galerkin finite element method for a linear system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion equations in the energy and balanced norms

<abstract><p>This paper introduces a weak Galerkin finite element method for a system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion problems. The proposed method is independent of parameter and uses piecewise discontinuous polynomials on interior of each element and constant on the boundary of each element. By the Schur complement technique, the interior unknowns can be locally efficiently eliminated from the resulting linear system, and the degrees of freedom of the proposed method are comparable with the classical FEM. It has been reported that the energy norm is not adequate for singularly perturbed reaction-diffusion problems since it can not efficiently reflect the behaviour of the boundary layer parts when the diffusion coefficient is very small. For the first time, the error estimates in the balanced norm has been presented for a system of coupled singularly perturbed problems when each equation has different parameter. Optimal and uniform error estimates have been established in the energy and balanced norm on an uniform Shishkin mesh. Finally, we carry out various numerical experiments to verify the theoretical findings.</p></abstract>

Error analysis of a weak Galerkin finite element method for two-parameter singularly perturbed differential equations in the energy and balanced norms

A weak Galerkin finite element method is proposed for solving singularly perturbed problems with two parameters. A robust uniform optimal convergence has been proved in the corresponding energy and a stronger balanced norms using piecewise higher order discontinuous polynomials on a piecewise uniform Shishkin mesh. Finally, we give some numerical experiments to support theoretical results.

A discontinuous least squares finite element method for the Helmholtz equation

AbstractWe propose a discontinuous least squares finite element method for solving the Helmholtz equation. The method is based on the norm least squares functional with the weak imposition of the continuity across the interior faces as well as the boundary conditions. We minimize the functional over the discontinuous polynomial spaces to seek numerical solutions. The wavenumber explicit error estimates to our method are established. The optimal convergence rate in the energy norm with respect to a fixed wavenumber is attained. The least squares functional can naturally serve as a posteriori estimator in the h‐adaptive procedure. It is convenient to implement the code due to the usage of discontinuous elements. Numerical results in two and three dimensions are presented to verify the error estimates.

Conforming virtual element approximations of the two-dimensional Stokes problem

The virtual element method (VEM) is a Galerkin approximation method that extends the finite element method to polytopal meshes. In this paper, we present two different conforming virtual element formulations for the numerical approximation of the Stokes problem that work on polygonal meshes. The velocity vector field is approximated in the virtual element spaces of the two formulations, while the pressure variable is approximated through discontinuous polynomials. Both formulations are inf-sup stable and convergent with optimal convergence rates in the L2 and energy norm. We assess the effectiveness of these numerical approximations by investigating their behavior on a representative benchmark problem. The observed convergence rates are in accordance with the theoretical expectations and a weak form of the zero-divergence constraint is satisfied at the machine precision level.

Discontinuous Galerkin approximations to second-kind Volterra integral equations with weakly singular kernel

The discontinuous Galerkin (DG) method is employed to solve second-kind Volterra integral equations (VIEs) with weakly singular kernels. It is proved that the quadrature DG (QDG) method obtained from the DG method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the piecewise discontinuous polynomial collocation method. The convergence theory is established for VIEs for both uniform and graded meshes. Some numerical experiments are given to illustrate the obtained theoretical results.

Analysis of an unfitted mixed finite element method for a class of quasi-Newtonian Stokes flow

We propose and analyze an unfitted method for a dual-dual mixed formulation of a class of Stokes models with variable viscosity depending on the velocity gradient, in which the pseudoestress, the velocity and its gradient are the main unknowns. On a fluid domain Ω with curved boundary Γ we consider a Dirichlet boundary condition and employ an approach previously applied to the Stokes equations with constant viscosity, which consists of approximating Ω by a polyhedral computational subdomain Ωh, not necessarily fitting Ω, where a Galerkin method is applied to compute solution. Furthermore, to approximate the Dirichlet data on the computational boundary Γh, we make use of a transferring technique based on integrating the discrete velocity gradient. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas of order k≥0 for the pseudostress, and discontinuous polynomials of degree k for the velocity and its gradient. For the a priori error analysis we provide suitable assumptions on the mesh near the boundary Γ ensuring that the associated Galerkin scheme is well-posed and optimally convergent with O(hk+1). Next, for the case when Γh is taken as a piecewise linear interpolation of Γ, we develop a reliable and quasi-efficient residual-based a posteriori error estimator. Numerical experiments verify our analysis and illustrate the performance of the associated a posteriori error indicator.

A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

<p style='text-indent:20px;'>The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. We numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular meshes, where the method coincides with the <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb{P}}_{{1}}-{\mathbb{P}}_{{0}} $\end{document}</tex-math></inline-formula> Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.</p>

Singularly Perturbed Reaction\u2013Diffusion Problems as First order systems

We consider a singularly perturbed reaction diffusion problem as a first order two-by-two system. Using piecewise discontinuous polynomials for the first component and H_{{{,mathrm{{div}},}}}-conforming elements for the second component we provide a convergence analysis on layer adapted meshes and an optimal convergence order in a balanced norm that is comparable with a balanced H^2-norm for the second order formulation.

A discontinuous Galerkin method for a diffuse-interface model of immiscible two-phase flows with soluble surfactant

A numerical method using discontinuous polynomial approximations is formulated for solving a phase-field model of two immiscible fluids with a soluble surfactant. The proposed scheme is shown to decay the total free Helmholtz energy at the discrete level, which is consistent with the continuous model dynamics. The scheme recovers the Langmuir adsorption isotherms at equilibrium. Simulations of spinodal decomposition, flow through a cylinder and flow through a sequence of pore throats show the dynamics of the flow with and without surfactant. Finally the numerical method is used to simulate fluid flows in the pore space of Berea sandstone obtained by micro-CT imaging.

NewH(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

This work proposes a family of multiscale hybrid-mixed methods for the two-dimensional linear elasticity problem on general polygonal meshes. The new methods approximate displacement, stress, and rotation using two-scale discretizations. The first scale level setting consists of approximating the traction variable (Lagrange multiplier) in discontinuous polynomial spaces, and of computing elementwise rigid body modes. In the second level, the methods are made effective by solving completely independent local boundary Neumann elasticity problems written in a mixed form with weak symmetry enforcedviathe rotation multiplier. Since the finite-dimensional space for the traction variable constraints the local stress approximations, the discrete stress field lies in theH(div) space globally and stays in local equilibrium with external forces. We propose different choices to approximate local problems based on pairs of finite element spaces defined on affine second-level meshes. Those choices generate the family of multiscale finite element methods for which stability and convergence are proved in a unified framework. Notably, we prove that the methods are optimal and high-order convergent in the natural norms. Also, it emerges that the approximate displacement and stress divergence are super-convergent in theL2-norm. Numerical verifications assess theoretical results and highlight the high precision of the new methods on coarse meshes for multilayered heterogeneous material problems.

A discontinuous finite element approximation to singular Lane-Emden type equations

In this article, we develop the local discontinuous Galerkin finite element method for the numerical approximations of a class of singular second-order ordinary differential equations known as the Lane-Emden type equations equipped with initial or boundary conditions. These equations have been considered via different models that naturally appear for example in several phenomena in astrophysical science. By converting the governing equations into a first-order systems of differential equations, the approximate solution is sought in a piecewise discontinuous polynomial space while the natural upwind fluxes are used at element interfaces. The existence-uniqueness of the weak formulation is provided and the numerical stability of the method in the L∞ norm is established. Five illustrative test problems are given to demonstrate the applicability and validity of the scheme. Comparisons between the numerical results of the proposed method with existing results are carried out in order to show that the new approximation algorithm provides accurate solutions even near the singular point.