Discontinuous Piecewise Polynomials Research Articles

A priori and a posteriori error analysis of a mixed DG method for the three-field quasi-Newtonian Stokes flow

Abstract In this paper we propose and analyse a new mixed-type DG method for the three-field quasi-Newtonian Stokes flow. The scheme is based on the introduction of the stress and strain tensor as further unknowns as well as the elimination of the pressure variable by means of the incompressibility constraint. As such, the resulting system involves three unknowns: the stress, the strain tensor and the velocity. All these three unknowns are approximated using discontinuous piecewise polynomials, which offers flexibility for enforcing the symmetry of the stress and the strain tensor. The unique solvability and a comprehensive convergence error analysis for all the variables are performed. All the variables are proved to converge optimally. Adaptive mesh refinement guided by a posteriori error estimator is computationally efficient, especially for problems involving singularity. In line of this mechanism we derive a residual-type a posteriori error estimator, which constitutes the second main contribution of the paper. In particular, we employ the elliptic reconstruction in conjunction with the Helmholtz decomposition to derive the a posteriori error estimator, which avoids using the averaging operator. Several numerical experiments are carried out to verify the theoretical findings.

A discontinuous piecewise polynomial generalized moving least squares scheme for robust finite element analysis on arbitrary grids

A discontinuous piecewise polynomial generalized moving least squares scheme for robust finite element analysis on arbitrary grids

Sparse grid time-discontinuous Galerkin method with streamline diffusion for transport equations

High-dimensional transport equations frequently occur in science and engineering. Computing their numerical solution, however, is challenging due to its high dimensionality. In this work we develop an algorithm to efficiently solve the transport equation in moderately complex geometrical domains using a Galerkin method stabilized by streamline diffusion. The ansatz spaces are a tensor product of a sparse grid in space and discontinuous piecewise polynomials in time. Here, the sparse grid is constructed upon nested multilevel finite element spaces to provide geometric flexibility. This results in an implicit time-stepping scheme which we prove to be stable and convergent. If the solution has additional mixed regularity, the convergence of a 2d-dimensional problem equals that of a d-dimensional one up to logarithmic factors. For the implementation, we rely on the representation of sparse grids as a sum of anisotropic full grid spaces. This enables us to store the functions and to carry out the computations on a sequence regular full grids exploiting the tensor product structure of the ansatz spaces. In this way existing finite element libraries and GPU acceleration can be used. The combination technique is used as a preconditioner for an iterative scheme to solve the transport equation on the sequence of time strips. Numerical tests show that the method works well for problems in up to six dimensions. Finally, the method is also used as a building block to solve nonlinear Vlasov-Poisson equations.

A five-field mixed formulation for stationary magnetohydrodynamic flows in porous media

We introduce and analyze a new mixed variational formulation for a stationary magnetohydrodynamic flows in porous media problem, whose governing equations are given by the steady Brinkman–Forchheimer equations coupled with the Maxwell equations. Besides the velocity, magnetic field and a Lagrange multiplier associated to the divergence-free condition of the magnetic field, a convenient translation of the velocity gradient and the pseudostress tensor are introduced as further unknowns. As a consequence, we obtain a five-field Banach spaces-based mixed variational formulation, where the aforementioned variables are the main unknowns of the system. The resulting mixed scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and a sufficiently small data assumption, are applied to prove the unique solvability of the continuous and discrete systems. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. The finite element discretization involves Raviart–Thomas elements of order k≥0 for the pseudostress tensor, discontinuous piecewise polynomial elements of degree k for the velocity and the translation of the velocity gradient, Nédélec elements of degree k for the magnetic field and Lagrange elements of degree k+1 for the associated Lagrange multiplier. Stability, convergence, and optimal a priori error estimates for the associated Galerkin scheme are obtained. Numerical tests illustrate the theoretical results.

Stabilized cut discontinuous Galerkin methods for advection–reaction problems on surfaces

We develop a novel cut discontinuous Galerkin (CutDG) method for stationary advection–reaction problems on surfaces embedded in Rd. The CutDG method is based on embedding the surface into a full-dimensional background mesh and using the associated discontinuous piecewise polynomials of order k as test and trial functions. As the surface can cut through the mesh in an arbitrary fashion, we design a suitable stabilization that enables us to establish inf-sup stability, a priori error estimates, and condition number estimates using an augmented streamline-diffusion norm. The resulting CutDG formulation is geometrically robust in the sense that all derived theoretical results hold with constants independent of any particular cut configuration. Numerical examples support our theoretical findings.

Dual least‐squares finite element method with stabilization

AbstractThe LL*‐method is a least‐squares finite element approach producing an approximation by solving dual problem corresponding to the given partial differential equations. Due to the unique structure of LL* approximation, it has advantages if the problem has low regularities and when L2‐approximation needs to be established. As a drawback, piecewise polynomial type approximation often generates artifacts such as spurious oscillations near where shocks or discontinuities occur in solution. Allowing discontinuous piecewise polynomial approximation in LL* seems to exacerbate this trouble. This paper presents a stabilized LL*‐method that is designed to effectively reduce these oscillatory behavior. The consistency and error convergence of proposed method are analyzed and numerical examinations are performed.

A vorticity-based mixed formulation for the unsteady Brinkman–Forchheimer equations

We propose and analyze an augmented mixed formulation for the time-dependent Brinkman–Forchheimer equations written in terms of vorticity, velocity and pressure. The weak formulation is based on the introduction of suitable least squares terms arising from the incompressibility condition and the constitutive equation relating the vorticity and velocity. We establish existence and uniqueness of a solution to the weak formulation, and derive the corresponding stability bounds, employing classical results on nonlinear monotone operators. We then propose a semidiscrete continuous-in-time approximation based on stable Stokes elements for the velocity and pressure, and continuous or discontinuous piecewise polynomial spaces for the vorticity. In addition, by means of the backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove well-posedness and derive the stability bounds for both schemes, and establish the corresponding error estimates. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.

Analysis of a new mixed FEM for stationary incompressible magneto-hydrodynamics

In this paper we propose and analyze a new mixed finite element method for a stationary magneto-hydrodynamic (MHD) model. The method is based on the utilization of a new dual-mixed formulation recently introduced for the Navier-Stokes problem, which is coupled with a classical primal formulation for the Maxwell equations. The latter implies that the velocity and a pseudostress tensor relating the velocity gradient with the convective term for the hydrodynamic equations, together with the magnetic field and a Lagrange multiplier related with the divergence-free property of the magnetic field, become the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree k for the aforementioned pseudostress tensor, discontinuous piecewise polynomial elements of degree k for the velocity, Nédélec elements of degree k for the magnetic field and Lagrange elements of degree k for the associated Lagrange multiplier. The analysis of the continuous and discrete problems are carried out by means of the Lax–Milgram lemma, the Banach–Nečas–Babuška and Banach fixed-point theorems, under a sufficiently small data assumption. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. We also develop an a priori error analysis and show that the proposed finite element method is optimal convergent. Finally, some numerical results illustrating the good performance of the method are provided.

A weak Galerkin finite element method for Allen–Cahn equation with a nonuniform two‐step backward differentiation formula scheme

AbstractIn this article, we consider a weak Galerkin finite element method and a nonuniform two‐step backward differentiation formula scheme for solving the Allen–Cahn equation. A discrete weak gradient operator on discontinuous piecewise polynomials is used in the numerical scheme. It is well‐known that the Allen–Cahn equations have a nonlinear stability property, that is, the free‐energy functional decreases with respect to time. A corresponding energy stability analysis of the discretized system is developed. Optimal order of L2‐norm error estimates are derived. Numerical experiments are preformed to support the theoretical results.

Limit Cycles of a Class of Discontinuous Piecewise Differential Systems Separated by the Curve y = xn Via Averaging Theory

Recently there is increasing interest in studying the limit cycles of the piecewise differential systems due to their many applications. In this paper we prove that the linear system [Formula: see text], [Formula: see text], can produce at most seven crossing limit cycles for [Formula: see text] using the averaging theory of first order, where the bounds [Formula: see text] for [Formula: see text] even and the bounds [Formula: see text] for [Formula: see text] odd are reachable, when it is perturbed by discontinuous piecewise polynomials formed by two pieces separated by the curve [Formula: see text] ([Formula: see text]), and having in each piece a quadratic polynomial differential system. Using the averaging theory of second order the perturbed system can be chosen in such way that it has [Formula: see text] or [Formula: see text] crossing limit cycles for [Formula: see text] even and, furthermore, under a particular condition we prove that the number of crossing limit cycles does not exceed [Formula: see text] (resp., [Formula: see text]) for [Formula: see text] even (resp., [Formula: see text] even). The averaging theory of second order produces the same number of crossing limit cycles as the averaging theory of first order if [Formula: see text] is odd. The main tools for proving our results are the new averaging theory developed for studying the crossing limit cycles of the discontinuous piecewise differential systems, and the theory for studying the zeros of a function using the extended Chebyshev systems.

Discontinuous piecewise polynomial collocation methods for integral-algebraic equations of Hessenberg type

Discontinuous piecewise polynomial collocation methods for integral-algebraic equations of Hessenberg type

Robust A Posteriori Error Analysis for Rotation-Based Formulations of the Elasticity/Poroelasticity Coupling

We develop the a posteriori error analysis of three mixed finite element formulations for rotation-based equations in elasticity, poroelasticity, and interfacial elasticity-poroelasticity. The discretizations use $H^1$-conforming finite elements of degree $k+1$ for displacement and fluid pressure, and discontinuous piecewise polynomials of degree $k$ for rotation vector, total pressure, and elastic pressure. Residual-based estimators are constructed, and upper and lower bounds (up to data oscillations) for all global estimators are rigorously derived. The methods are all robust with respect to the model parameters (in particular, the Lamé constants); they are valid in 2D and 3D, and also for arbitrary polynomial degree $k\geq 0$. The error behavior predicted by the theoretical analysis is then demonstrated numerically on a set of computational examples including different geometries on which we perform adaptive mesh refinement guided by the a posteriori error estimators.

Regularity results and numerical solution by the discontinuous Galerkin method to semilinear parabolic initial boundary value problems with nonlinear Newton boundary conditions in a polygonal space-time cylinder

Abstract In this note we consider a parabolic evolution equation in a polygonal space-time cylinder. We show, that the elliptic part is given by a m-accretive mapping from L q (Ω) → L q (Ω). Therefore we can apply the theory of nonlinear semigroups in Banach spaces in order to get regularity results in time and space. The second part of the paper deals with the numerical solution of the problem. It is dedicated to the application of the space-time discontinuous Galerkin method (STDGM). It means that both in space as well as in time discontinuous piecewise polynomial approximations of the solution are used. We concentrate to the theoretical analysis of the error estimation.

A stabilizer-free C0 weak Galerkin method for the biharmonic equations

In this article, we present and analyze a stabilizer-free C0 weak Galerkin (SF-C0WG) method for solving the biharmonic problem. The SF-C0WG method is formulated in terms of cell unknowns which are C0 continuous piecewise polynomials of degree k + 2 with k ≽ 0 and in terms of face unknowns which are discontinuous piecewise polynomials of degree k + 1. The formulation of this SF-C0WG method is without the stabilized or penalty term and is as simple as the C1 conforming finite element scheme of the biharmonic problem. Optimal order error estimates in a discrete H2-like norm and the H1 norm for k ≽ 0 are established for the corresponding WG finite element solutions. Error estimates in the L2 norm are also derived with an optimal order of convergence for k > 0 and sub-optimal order of convergence for k = 0. Numerical experiments are shown to confirm the theoretical results.

A three-field Banach spaces-based mixed formulation for the unsteady Brinkman–Forchheimer equations

We propose and analyze a new mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Besides the velocity, our approach introduces the velocity gradient and a pseudostress tensor as further unknowns. As a consequence, we obtain a three-field Banach spaces-based mixed variational formulation, where the aforementioned variables are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation, and derive the corresponding stability bounds, employing classical results on nonlinear monotone operators. We then propose a semidiscrete continuous-in-time approximation on simplicial grids based on the Raviart–Thomas elements of degree k≥0 for the pseudostress tensor and discontinuous piecewise polynomials of degree k for the velocity and the velocity gradient. In addition, by means of the backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove well-posedness and derive the stability bounds for both schemes, and under a quasi-uniformity assumption on the mesh, we establish the corresponding error estimates. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.

Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations

The aim of this paper is to carry out an improved analysis of the convergence of the Nyström and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree ⩽r−1, we obtain convergence order 2r for degenerate kernel and Nyström methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are 3r+1 and 4r, respectively. Moreover, we show that the optimal convergence order 4r is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples.

A C 0-conforming DG finite element method for biharmonic equations on triangle/tetrahedron

Abstract A C 0-conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the biharmonic equation. The first strong gradient of C 0 finite element functions is a vector of discontinuous piecewise polynomials. The second gradient is the weak gradient of discontinuous piecewise polynomials. This method, by its name, uses nonconforming (non C 1) approximations and keeps simple formulation of conforming finite element methods without any stabilizers. Optimal order error estimates in both a discrete H 2-norm and the L 2-norm are established for the corresponding finite element solutions. Numerical results are presented to confirm the theory of convergence.

Equivalence of local-best and global-best approximations in H(curl)

We derive results on equivalence of piecewise polynomial approximations of a given function in the Sobolev space $${\varvec{H}}{\mathrm{(curl)}}$$ . We namely show that the global-best approximation of a given $${\varvec{H}}{\mathrm{(curl)}}$$ function in a $${\varvec{H}}{\mathrm{(curl)}}$$ -conforming piecewise polynomial space imposing the continuity of the tangential trace can be bounded above and below by the Hilbertian sum of the respective local approximations from the elementwise spaces without any inter-element continuity requirement. In other words, the approximation of a $${\varvec{H}}{\mathrm{(curl)}}$$ function by tangential-trace-continuous and discontinuous piecewise polynomials has comparable precision. We consider approximations of the curl of the target function in the $${\varvec{L}} ^2$$ -norm, as well as approximations of the target function in the $${\varvec{L}} ^2$$ -norm with a constraint on the curl; in the latter case, the constraint is removed in the local approximations. These best-approximation localizations hold under the minimal $${\varvec{H}}{\mathrm{(curl)}}$$ regularity, on arbitrary shape-regular tetrahedral meshes, and include imposition of conditions on a part of the boundary. They extend to the $${\varvec{H}}{\mathrm{(curl)}}$$ context some recent results from the $$H^1$$ and $${\varvec{H}}{\mathrm{(div)}}$$ spaces and have direct applications to a priori and a posteriori error analysis of numerical discretizations related to the $${\varvec{H}}{\mathrm{(curl)}}$$ space, namely Maxwell’s equations.

An Lpspaces-based formulation yielding a new fully mixed finite element method for the coupled Darcy and heat equations

AbstractIn this work we present and analyse a new fully mixed finite element method for the nonlinear problem given by the coupling of the Darcy and heat equations. Besides the velocity, pressure and temperature variables of the fluid, our approach is based on the introduction of the pseudoheat flux as a further unknown. As a consequence of it, and due to the convective term involving the velocity and the temperature, we arrive at saddle point-type schemes in Banach spaces for both equations. In particular, and as suggested by the solvability of a related Neumann problem to be employed in the analysis, we need to make convenient choices of the Lebesgue and ${\textrm {H}}(div)$-type spaces to which the unknowns and test functions belong. The resulting coupled formulation is then written equivalently as a fixed-point operator, so that the classical Banach theorem, combined with the corresponding Babuška–Brezzi theory, the Banach–Nečas–Babuška theorem, suitable operators mapping Lebesgue spaces into themselves, regularity assumptions and the aforementioned Neumann problem, are employed to establish the unique solvability of the continuous formulation. Under standard hypotheses satisfied by generic finite element subspaces, the associated Galerkin scheme is analysed similarly and the Brouwer theorem yields existence of a solution. The respective a priori error analysis is also derived. Then, Raviart–Thomas elements of order $k\ge 0$ for the pseudoheat and the velocity and discontinuous piecewise polynomials of degree $\le k$ for the pressure and the temperature are shown to satisfy those hypotheses in the two-dimensional case. Several numerical examples illustrating the performance and convergence of the method are reported, including an application into the equivalent problem of miscible displacement in porous media.

On the 16th Hilbert Problem for Discontinuous Piecewise Polynomial Hamiltonian Systems

In this paper we study the maximum number of limit cycles of the discontinuous piecewise differential systems with two zones separated by the straight line \(y=0\), in \(y\ge 0\) there is a polynomial Hamiltonian system of degree m, and in \(y\le 0\) there is a polynomial Hamiltonian system of degree n. First for this class of discontinuous piecewise polynomial Hamiltonian systems, which are perturbation of a linear center, we provide a sharp upper bound for the maximum number of the limit cycles that can bifurcate from the periodic orbits of the linear center using the averaging theory up to any order. After for the general discontinuous piecewise polynomial Hamiltonian systems we also give an upper bound for their maximum number of limit cycles in function of m and n. Moreover, this upper bound is reached for some degrees of m and n.