Continuous Piecewise Polynomial Research Articles

Construction of Supplemental Functions for Direct Serendipity and Mixed Finite Elements on Polygons

New families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons were recently defined by the authors. The finite elements of index r are H1 and H(div) conforming, respectively, and approximate optimally to order r+1 while using the minimal number of degrees of freedom. The shape function space consists of the full set of polynomials defined directly on the element and augmented with a space of supplemental functions. The supplemental functions were constructed as rational functions, which can be difficult to integrate accurately using numerical quadrature rules when the index is high. This can result in a loss of accuracy in certain cases. In this work, we propose alternative ways to construct the supplemental functions on the element as continuous piecewise polynomials. One approach results in supplemental functions that are in Hp for any p≥1. We prove the optimal approximation property for these new finite elements. We also perform numerical tests on them, comparing results for the original supplemental functions and the various alternatives. The new piecewise polynomial supplements can be integrated accurately, and therefore show better robustness with respect to the underlying meshes used.

An augmented mixed FEM for the convective Brinkman–Forchheimer problem: a priori and a posteriori error analysis

We propose and analyze an augmented mixed finite element method for the pseudostress–velocity formulation of the stationary convective Brinkman–Forchheimer problem in Rd,d∈{2,3}. Since the convective and Forchheimer terms forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram theorem, allow to prove the unique solvability of the continuous problem. The finite element discretization involves Raviart–Thomas spaces of order k≥0 for the pseudostress tensor and continuous piecewise polynomials of degree ≤k+1 for the velocity. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. In addition, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the form involved, a suitable assumption on the data, a stable Helmholtz decomposition, and the local approximation properties of the Clément and Raviart–Thomas operators. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, some numerical examples illustrating the performance of the mixed finite element method, confirming the theoretical rate of convergence and the properties of the estimators, and showing the behavior of the associated adaptive algorithms, are reported. In particular, the case of flow through a 2D porous media with fracture networks is considered.

Analysis of a fractional viscoelastic Euler-Bernoulli beam and identification of its piecewise continuous polynomial order

Analysis of a fractional viscoelastic Euler-Bernoulli beam and identification of its piecewise continuous polynomial order

Revisiting the robustness of the multiscale hybrid-mixed method: The face-based strategy

This work proposes a new finite element for the multiscale hybrid-mixed method (MHM) applied to the Poisson equation with highly oscillatory coefficients. Unlike the original MHM method, multiscale bases are the solution to local Neumann problems driven by piecewise continuous polynomial interpolation on the skeleton faces of the macroscale mesh. As a result, we prove the optimal convergence of MHM by refining the face partition and leaving the mesh of macroelements fixed. This property allows the MHM method to be resonance free under the usual assumptions of local regularity. The numerical analysis of the method also revisits and complements the original approach proposed by D. Paredes, F. Valentin and H. Versieux (2017). Numerical experiments assess the new theoretical results.

Central part interpolation schemes for fractional differential equations

We consider a two-point boundary value problem for fractional integro-differential equations with weakly singular kernels. Using an integral equation reformulation of the underlying problem, a collocation method based on the central part interpolation by continuous piecewise polynomials on the uniform grid is constructed and analysed. Optimal convergence order of the proposed method is established and confirmed by numerical experiments.

Построение С1-гладких кусочно-квадратичных функций при решении краевых задач уравнений 4-го порядка на треугольной сетке

In this paper, we present one approach to constructing continuously differentiable piecewise-quadratic functions on a triangular grid, based on smoothing a piecewise-linear function in the vicinity of edges and triangulation nodes. In the works [8], [7], the issues of approximation of the functional (1) in triangular grids and the convergence of the variational method for solving the boundary value problem of the equation (2) were studied. However, in the numerical solution there are difficulties associated with the construction of continuously differentiable piecewise polynomial functions on triangulations. In particular, their construction requires solving large systems of equations at each step of the variational method. When trying to get by with only continuous piecewise polynomial functions, we got a negative result (divergence of approximate solutions was found) [9]. In this paper, we circumvent the difficulties that arise — we indicate a method for constructing piecewise polynomial functions that have continuous partial derivatives. With the help of this class of functions, we have obtained formulas for approximating the functional (1) and tested the method on the example of a biharmonic equation.

A real‐time decoupling trajectory planning method for on‐road autonomous driving

AbstractTrajectory planning is a core technology for autonomous vehicle directlyreflecting the driving safety and efficiency. In this paper, a spatial‐speed decoupled planning method is studied for real‐time trajectory generation in the on‐road environment. The proposed approach mainly includes two parts: optimization‐based lateral planning and selection‐based longitudinal speed planning. The optimization‐based lateral planning is employed to generate an optimal collision‐free and smooth path by solving a quadratic programming problem. Specifically, the lateral planning first constructs a safe corridor by integrating obstacles and initial risky corridor together in the Frénet frame. The safe corridor is useful for quadratic programming problem formulation. The selecting‐based longitudinal speed planning is proposed to generate a suitable and continuous velocity trajectory. The novel speed selector considering four cases is designed to select a more suitable reference velocity in different lane situations. The continuous velocity trajectory is obtained by solving piecewise continuous quartic polynomial. As a result, the combination of the spatial path and the velocity trajectory is the final planning result. The proposed algorithm is tested extensively in a simulation environment. Experimental results demonstrate that the proposed algorithm has good real‐time property and practical validity.

Error profile for discontinuous Galerkin time stepping of parabolic PDEs

We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most r − 1 in t, for r ≥ 1 and with maximum step size k. It is well known that the spatial L2-norm of the DG error is of optimal order kr globally in time, and is, for r ≥ 2, superconvergent of order k2r− 1 at the nodes. We show that on the n th subinterval (tn− 1,tn), the dominant term in the DG error is proportional to the local right Radau polynomial of degree r. This error profile implies that the DG error is of order kr+ 1 at the right-hand Gauss–Radau quadrature points in each interval. We show that the norm of the jump in the DG solution at the left end point tn− 1 provides an accurate a posteriori estimate for the maximum error over the subinterval (tn− 1,tn). Furthermore, a simple post-processing step yields a continuous piecewise polynomial of degree r with the optimal global convergence rate of order kr+ 1. We illustrate these results with some numerical experiments.

Immersed boundary Mindlin-Reissner 3D shell element for modeling isotropic and laminated composite shells

An immersed boundary 3D shell element is presented here that is based on Mindlin-Reissner shell theory assumptions and uses quadratic B-spline approximation for the solution. It enables mesh independent analysis wherein the surface representing the shell geometry is defined independently and not by the mesh. The shell geometry is immersed in a uniform background mesh whose elements are 3D B-spline shell elements. By making the geometric model independent of the mesh, the typical difficulties associated with mesh generation can be avoided. The quadratic 3D B-spline shell elements in the mesh can represent the displacement field as a tangent continuous piece-wise polynomial approximation. The nodes of the element have three translational and no rotational degrees of freedom. The element is formulated under small-deformation and plane stress assumptions. Constructing the element based on shell theory enables the use of effective properties to model thin fiber reinforced laminated composite shell-like structures. The element formulation and the mesh independent approach are validated against a series of common shell element benchmark problems. • Immersed boundary 3D B-spline element based on Mindlin-Reissner shell theory assumptions. • Mesh independent analysis where geometry is embedded in a uniform background mesh. • The iso-parametric quadratic B-spline shell elements are used for C 1 continuous approximation. • Step boundary method (SBM) is used for applying boundary conditions. • Immersed boundary FEM for shells is validated for isotropic and composite shells.

Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree N inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor. The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme.

Central part interpolation schemes for a class of fractional initial value problems

We consider an initial value problem for linear fractional integro-differential equations with weakly singular kernels. Using an integral equation reformulation of the underlying problem, a collocation method based on the central part interpolation by continuous piecewise polynomials on the uniform grid is constructed and analysed. Optimal convergence order of the proposed method is established and confirmed by numerical experiments.

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 divergence-free finite element method for the Stokes problem with boundary correction

Abstract This paper constructs and analyzes a boundary correction finite element method for the Stokes problem based on the Scott–Vogelius pair on Clough–Tocher splits. The velocity space consists of continuous piecewise polynomials of degree k, and the pressure space consists of piecewise polynomials of degree (k – 1) without continuity constraints. A Lagrange multiplier space that consists of continuous piecewise polynomials with respect to the boundary partition is introduced to enforce boundary conditions and to mitigate the lack of pressure-robustness. We prove several inf-sup conditions, leading to the well-posedness of the method. In addition, we show that the method converges with optimal order and the velocity approximation is divergence-free.

Piecewise discretization of monodromy operators of delay equations on adapted meshes

<p style='text-indent:20px;'>Periodic solutions of delay equations are usually approximated as continuous piecewise polynomials on meshes adapted to the solutions' profile. In practical computations this affects the regularity of the (coefficients of the) linearized system and, in turn, the effectiveness of assessing local stability by approximating the Floquet multipliers. To overcome this problem when computing multipliers by collocation, the discretization grid should include the piecewise adapted mesh of the computed periodic solution. By introducing a piecewise version of existing pseudospectral techniques, we explain why and show experimentally that this choice is essential in presence of either strong mesh adaptation or nontrivial multipliers whose eigenfunctions' profile is unrelated to that of the periodic solution.</p>

Hidden positivity and a new approach to numerical computation of Hausdorff dimension: higher order methods

In [14], the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend this approach to incorporate high order approximation methods. We again rely on the fact that we can associate to the IFS a parametrized family of positive, linear, Perron-Frobenius operators $L_s$, an idea known in varying degrees of generality for many years. Although $L_s$ is not compact in the setting we consider, it possesses a strictly positive $C^m$ eigenfunction $v_s$ with eigenvalue $R(L_s)$ for arbitrary $m$ and all other points $z$ in the spectrum of $L_s$ satisfy $|z| \le b$ for some constant $b < R(L_s)$. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value $s=s_*$ for which $R(L_s) =1$. This eigenvalue problem is then approximated by a collocation method at the extended Chebyshev points of each subinterval using continuous piecewise polynomials of arbitrary degree $r$. Using an extension of the Perron theory of positive matrices to matrices that map a cone $K$ to its interior and explicit a priori bounds on the derivatives of the strictly positive eigenfunction $v_s$, we give rigorous upper and lower bounds for the Hausdorff dimension $s_*$, and these bounds converge rapidly to $s_*$ as the mesh size decreases and/or the polynomial degree increases.

Error analysis for a vorticity/Bernoulli pressure formulation for the Oseen equations

Abstract A variational formulation is analysed for the Oseen equations written in terms of vorticity and Bernoulli pressure. The velocity is fully decoupled using the momentum balance equation, and it is later recovered by a post-process. A finite element method is also proposed, consisting in equal-order Nédélec finite elements and piecewise continuous polynomials for the vorticity and the Bernoulli pressure, respectively. The a priori error analysis is carried out in the L2-norm for vorticity, pressure, and velocity; under a smallness assumption either on the convecting velocity, or on the mesh parameter. Furthermore, an a posteriori error estimator is designed and its robustness and efficiency are studied using weighted norms. Finally, a set of numerical examples in 2D and 3D is given, where the error indicator serves to guide adaptive mesh refinement. These tests illustrate the behaviour of the new formulation in typical flow conditions, and also confirm the theoretical findings.

On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

We provide an endpoint stability result for Scott-Zhang type operators in Besov spaces. For globally continuous piecewise polynomials these are bounded from H3/2 into B3/22,∞; for element wise polynomials these are bounded from H1/2 into B1/22,∞. As an application, we obtain a multilevel decomposition based on Scott-Zhang operators on a hierarchy of meshes generated by newest vertex bisection with equivalent norms up to (but excluding) the endpoint case. A local multilevel diagonal preconditioner for the fractional Laplacian on locally refined meshes with optimal eigenvalue bounds is presented.

Collocation methods for cordial Volterra integro-differential equations

This paper is concerned with collocation methods for cordial Volterra integro-differential equations (CVIDEs) with noncompact cordial operators. The existence, uniqueness and regularity of the exact solutions to CVIDEs are discussed, and a resolvent representation of the derivative of the exact solution is obtained. We approximate the exact solution by collocation in the space of continuous piecewise polynomials of degree m. The solvability of the collocation equations is proved for sufficiently small meshes diameter. It is shown that, if the solution is sufficiently smooth, the collocation solutions are convergent with global order of convergence m. Using an approach based on the resolvent formula, we prove that global superconvergence of order m+1 is attained with iterated collocation based on some special points. Some numerical examples are provided to verify the convergence results.

A fully‐mixed finite element method for the coupling of the Navier–Stokes and Darcy–Forchheimer equations

AbstractIn this work we present and analyze a fully‐mixed formulation for the nonlinear model given by the coupling of the Navier–Stokes and Darcy–Forchheimer equations with the Beavers–Joseph–Saffman condition on the interface. Our approach yields non‐Hilbertian normed spaces and a twofold saddle point structure for the corresponding operator equation. Furthermore, since the convective term in the Navier–Stokes equation forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well‐known Schauder and Banach theorems, combined with classical results on nonlinear monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. In particular, given an integer k ≥ 0, Raviart–Thomas spaces of order k, continuous piecewise polynomials of degree ≤k + 1 and piecewise polynomials of degree ≤k are employed in the fluid for approximating the pseudostress tensor, velocity and vorticity, respectively, whereas Raviart–Thomas spaces of order k and piecewise polynomials of degree ≤k for the velocity and pressure, constitute a feasible choice in the porous medium. A priori error estimates and associated rates of convergence are derived, and several numerical examples illustrating the good performance of the method are reported.

A model for incompressible fluids using finite element methods for the Brinkman problem

The partial differential equations for fluid flow dynamics based on the Brinkman equations, written in terms of velocity-vorticity and pressure in three dimensions, are essential for predicting climate, ocean currents, water flow in a pipe, the study of blood flow and any phenomenon involving incompressible fluids through porous media; having a significant impact in areas such as oceanographic engineering and biomedical sciences. This paper aims to study the Brinkman equations with homogeneous Dirichlet boundary are studied, the existence and uniqueness of solution at a continuous level through equivalence of problems is presented. It is discretized to approximate the solution using Nédélec finite elements and piecewise continuous polynomials to approximate vorticity and pressure. The velocity field is recovered, obtaining its a priori error estimation and order of convergence. As a result, ensuring a single prediction of the flow behavior of an incompressible fluid through porous media. Finally, a numerical example in 2D with the standard L2 is presented, confirming the theoretical analysis.