Piecewise Polynomial Spaces Research Articles

On interpolation spaces of piecewise polynomials on mixed meshes

We consider fractional Sobolev spaces Hθ, θ ∈ (0,1), on 2D domains and H1-conforming discretizations by globally continuous piecewise polynomials on a mesh consisting of shaperegular triangles and quadrilaterals. We prove that the norm obtained from interpolating between the discrete space equipped with the L2-norm on the one hand and the H1-norm on the other hand is equivalent to the corresponding continuous interpolation Sobolev norm, and the norm-equivalence constants are independent of meshsize and polynomial degree. This characterization of the Sobolev norm is then used to show an inverse inequality between H1 and Hθ.

Lower bounds for piecewise polynomial approximations of oscillatory functions

We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when the polynomial degree is fixed. These lower bounds, for example, apply when approximating solutions to Helmholtz plane wave scattering problem.

Discrete tensor product BGG sequences: Splines and finite elements

In this paper, we provide a systematic discretization of the Bernstein-Gelfand-Gelfand diagrams and complexes over cubical meshes in arbitrary dimension via the use of tensor product structures of one-dimensional piecewise-polynomial spaces, such as spline and finite element spaces. We demonstrate the construction of the Hessian, the elasticity, and div ⁡ div \operatorname {div}\operatorname {div} complexes as examples for our construction.

Polyhedral control-net splines for analysis

Generalizing tensor-product splines to smooth functions whose control nets can outline more general topological polyhedra, bi-cubic polyhedral-net splines form a first-order differentiable piecewise polynomial space. Each polyhedral control net node is associated with one bi-cubic function. A polyhedral control net admits grid-, star-, n-gon-, polar- and three types of T-junction configurations. Analogous to tensor-product splines, polyhedral-net splines can both model curved geometry and represent higher-order functions on the geometry. This paper explores the use of polyhedral-net splines for solving elliptic partial differential equations on curved smooth free-form surfaces without additional meshing.

APPROXIMATION OF WEAKLY SINGULAR NON-LINEAR VOLTERRA-URYSOHN INTEGRAL EQUATIONS BY PIECEWISE POLYNOMIAL PROJECTION METHODS BASED ON GRADED MESH

In this article, we address the approximation solution of VolterraUrysohn integral equations which involves weakly singular kernels. In order to get better convergence rates, projection methods namely Galerkin and multi Galerkin methods, along with their iterated versions are used in the space of piecewise polynomials subspaces based on the graded mesh. In addition, we compute the superconvergence results for the proposed integral equation and show that iterated Galerkin method outperforms Galerkin method in terms of order of convergence. Further, we demonstrate numerical examples to verify the proposed theoretical framework.

A cutFEM divergence–free discretization for the stokes problem

We construct and analyze a CutFEM discretization for the Stokes problem based on the Scott–Vogelius pair. The discrete piecewise polynomial spaces are defined on macro-element triangulations which are not fitted to the smooth physical domain. Boundary conditions are imposed via penalization through the help of a Nitsche-type discretization, whereas stability with respect to small and anisotropic cuts of the bulk elements is ensured by adding local ghost penalty stabilization terms. We show stability of the scheme as well as a divergence–free property of the discrete velocity outside an O(h) neighborhood of the boundary. To mitigate the error caused by the violation of the divergence–free condition, we introduce local grad–div stabilization. The error analysis shows that the grad–div parameter can scale like O(h−1), allowing a rather heavy penalty for the violation of mass conservation, while still ensuring optimal order error estimates.

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.

On the SAV‐DG method for a class of fourth order gradient flows

AbstractFor a class of fourth order gradient flow problems, integration of the scalar auxiliary variable (SAV) time discretization with the penalty‐free discontinuous Galerkin (DG) spatial discretization leads to SAV‐DG schemes. These schemes are linear and shown unconditionally energy stable. However, the reduced linear systems are rather expensive to solve due to the dense coefficient matrices. In this paper, we provide a procedure to pre‐evaluate the auxiliary variable in the piecewise polynomial space. As a result the computational complexity of reduces to when exploiting the conjugate gradient (CG) solver. This hybrid SAV‐DG method is more efficient and able to deliver satisfactory results of high accuracy. This was also compared with solving the full augmented system of the SAV‐DG schemes.

Computing discrete harmonic differential forms in a given cohomology class using finite element exterior calculus

Computational topology research of the past two decades has emphasized combinatorial techniques while numerical methods such as numerical linear algebra remain underutilized. While the combinatorial techniques have been very successful in diverse areas, for some applications, it is worth considering the numerical counterparts. We discuss one such application. Harmonic forms are elements of the kernel of the Hodge Laplacian operator and contain information about the topology of the manifold. If a particular cohomology class is chosen, the closed differential form with the smallest norm in that class is a harmonic form. We use these well-known facts to give an algorithm for solving the following problem: given a piecewise flat manifold simplicial complex (with or without boundary) and a closed piecewise polynomial differential form representing a cohomology class, find the discrete harmonic form in that cohomology class. We give a least squares based algorithm to solve this problem and show that the computed form satisfies the finite element exterior calculus (FEEC) equations for being a harmonic form. The piecewise polynomial spaces used are the spaces of trimmed polynomial forms, that is, arbitrary degree polynomial generalizations of Whitney forms which are used in FEEC. We also survey other methods for finding harmonic forms.

Convergent adaptive hybrid higher-order schemes for convex minimization

This paper proposes two convergent adaptive mesh-refining algorithms for the hybrid high-order method in convex minimization problems with two-sided p-growth. Examples include the p-Laplacian, an optimal design problem in topology optimization, and the convexified double-well problem. The hybrid high-order method utilizes a gradient reconstruction in the space of piecewise Raviart–Thomas finite element functions without stabilization on triangulations into simplices or in the space of piecewise polynomials with stabilization on polytopal meshes. The main results imply the convergence of the energy and, under further convexity properties, of the approximations of the primal resp. dual variable. Numerical experiments illustrate an efficient approximation of singular minimizers and improved convergence rates for higher polynomial degrees. Computer simulations provide striking numerical evidence that an adopted adaptive HHO algorithm can overcome the Lavrentiev gap phenomenon even with empirical higher convergence rates.

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.

Richardson extrapolation for the iterated Galerkin solution of Urysohn integral equations with Green's kernels

We consider a Urysohn integral operator with the kernel of the type of Green's function. For , a space of piecewise polynomials of degree with respect to a uniform partition is chosen to be the approximating space, and the projection is chosen to be the orthogonal projection. The iterated Galerkin method is applied to the integral equation . It is known that the order of convergence of the iterated Galerkin solution is r + 2 and is at the partition points. We obtain an asymptotic expansion of the iterated Galerkin solution at the partition points of the above Urysohn integral equation. Richardson extrapolation is used to improve the order of convergence. A numerical example is considered to illustrate our theoretical results.

Stable broken 𝐻(𝑐𝑢𝑟𝑙) polynomial extensions and 𝑝-robust a posteriori error estimates by broken patchwise equilibration for the curl–curl problem

We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizers over the broken H ( curl ) \boldsymbol H(\boldsymbol {\operatorname {curl}}) space with the same prescriptions. Our proofs are constructive and yield constants independent of the polynomial degree. We then detail the application of this result to the a posteriori error analysis of the curl–curl problem discretized with Nédélec finite elements of arbitrary order. The resulting estimators are reliable, locally efficient, polynomial-degree-robust, and inexpensive. They are constructed by a broken patchwise equilibration which, in particular, does not produce a globally H ( curl ) \boldsymbol H(\boldsymbol {\operatorname {curl}}) -conforming flux. The equilibration is only related to edge patches and can be realized without solutions of patch problems by a sweep through tetrahedra around every mesh edge. The error estimates become guaranteed when the regularity pick-up constant is explicitly known. Numerical experiments illustrate the theoretical findings.

L 1 -Multiscale Galerkin’s Scheme with Multilevel Augmentation Algorithm for Solving Time Fractional Burgers’ Equation

In this paper, we consider the initial boundary value problem of the time fractional Burgers equation. A fully discrete scheme is proposed for the time fractional nonlinear Burgers equation with time discretized by L 1 -type formula and space discretized by the multiscale Galerkin method. The optimal convergence orders reach O τ 2 − α + h r in the L 2 norm and O τ 2 − α + h r − 1 in the H 1 norm, respectively, in which τ is the time step size, h is the space step size, and r is the order of piecewise polynomial space. Then, a fast multilevel augmentation method (MAM) is developed for solving the nonlinear algebraic equations resulting from the fully discrete scheme at each time step. We show that the MAM preserves the optimal convergence orders, and the computational cost is greatly reduced. Numerical experiments are presented to verify the theoretical analysis, and comparisons between MAM and Newton’s method show the efficiency of our algorithm.

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.

An adaptive high-order piecewise polynomial based sparse grid collocation method with applications

This paper constructs adaptive sparse grid collocation method onto arbitrary order piecewise polynomial space. The sparse grid method is a popular technique for high dimensional problems, and the associated collocation method has been well studied in the literature. The contribution of this work is the introduction of a systematic framework for collocation onto high-order piecewise polynomial space that is allowed to be discontinuous. We consider both Lagrange and Hermite interpolation methods on nested collocation points. Our construction includes a wide range of function space, including those used in sparse grid continuous finite element method. Error estimates are provided, and the numerical results in function interpolation, integration and some benchmark problems in uncertainty quantification are used to compare different collocation schemes.

A discontinuous least squares finite element method for time-harmonic Maxwell equations

Abstract We propose and analyze a discontinuous least squares finite element method for solving the indefinite time-harmonic Maxwell equations. The scheme is based on the $L^2$ norm least squares functional with weak imposition of continuity across interior faces. We minimize the functional over the piecewise polynomial spaces to seek numerical solutions. The method is shown to be stable without any constraint on the mesh size. We prove the optimal convergence rate under the energy norm and sub-optimal convergence rate under the $L^2$ norm. Numerical results in two and three dimensions are presented to verify the error estimates.

Residual-Based A Posteriori Error Estimates for $hp$-Discontinuous Galerkin Discretizations of the Biharmonic Problem

We introduce a residual-based a posteriori error estimator for a novel hp-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of a C1-conforming piecewise polynomial space and is instead based on an elliptic reconstruction of the discrete solution to the H2 space and a generalised Helmholtz decomposition of the error. This is the first hp-version error estimator for the biharmonic problem in two and three dimensions. The practical behaviour of the estimator is investigated through numerical examples in two and three dimensions.

Asymptotic expansion of iterated Galerkin solution of Fredholm integral equations of the second kind with Green's kernel

We consider a Fredholm integral equation of the second kind with kernel of the type of Green’s function. Iterated Galerkin method is applied to such an integral equation. For r≥1, a space of piecewise polynomials of degree ≤r−1 with respect to a uniform partition is chosen to be the approximating space. We obtain an asymptotic expansion for the iterated Galerkin solution at the partition points. Richardson extrapolation is used to increase the order of convergence. A numerical example is considered to illustrate our theoretical results.

A nonconforming scheme with piecewise quasi three degree polynomial space to solve biharmonic problem

A new $$C^0$$ nonconforming quasi three degree element with 13 freedoms is introduced to solve biharmonic problem. The given finite element space consists of piecewise polynomial space $$P_3$$ and some bubble functions. Different from non- $$C^0$$ nonconforming scheme, a smoother discrete solution can be obtained by this method. Compared with the existed 16 freedoms finite element method, this scheme uses less freedoms. As the finite elements are not affine equivalent each other, the associated interpolating error estimation is technically proved by introducing another affine finite elements. With this space to solve biharmonic problem, the convergence analysis is demonstrated between true solution and discrete solution. Under a stronger hypothesis that true solution $$u\in H_0^2(\Omega )\cap H^4(\Omega )$$ , the scheme is of linear order convergence by the measurement of discrete norm $$\Vert \cdot \Vert _h$$ . Some numerical results are included to further illustrate the convergence analysis.