Piecewise Polynomial Spaces Research Articles

On a class of noncompact weakly singular Volterra integral equations: theory and application to fractional differential equations with variable coefficient

The purpose of this paper is twofold. We first carry out an analysis of a class of noncompact weakly singular Volterra integral equations whose kernels possess both an end-point and diagonal singularities. A numerical method based on piecewise collocation scheme on uniform meshes including its convergence analysis are presented. We will then indicate the usefulness of our main concerned equation in the numerical study of an initial-value problem for fractional differential equations with variable coefficient. As a result, we reformulate the linear fractional differential equations with Erdélyi–Kober derivative to a particular type of the underlying Volterra equations with weakly singular kernels. Under certain verifiable conditions on the coefficient, the existence and uniqueness results as well as the numerical solution of the resulting equation by spline collocation method on piecewise polynomial space are analyzed. The reliability and efficiency of this approach are finally demonstrated by some numerical experiments.

DISCRETE MODIFIED PROJECTION METHODS FOR URYSOHN INTEGRAL EQUATIONS WITH GREEN’S FUNCTION TYPE KERNELS

In the present paper we consider discrete versions of the modified projection methods for solving a Urysohn integral equation with a kernel of the type of Green’s function. For r ≥ 0, a space of piecewise polynomials of degree ≤ r with respect to an uniform partition is chosen to be the approximating space. We define a discrete orthogonal projection onto this space and replace the Urysohn integral operator by a Nyström approximation. The order of convergence which we obtain for the discrete version indicates the choice of numerical quadrature which preserves the orders of convergence in the continuous modified projection methods. Numerical results are given for a specific example.

Richardson extrapolation for the discrete iterated modified projection solution

Approximate solutions of Urysohn integral equations using projection methods involve integrals which need to be evaluated using a numerical quadrature formula. It gives rise to the discrete versions of the projection methods. For $r \geq 1,$ a space of piece-wise polynomials of degree $\leq r - 1$ with respect to an uniform partition is chosen to be the approximating space and the projection is chosen to be the interpolatory projection at $r$ Gauss points. Asymptotic expansion for the iterated modified projection solution is available in literature. In this paper, we obtain an asymptotic expansion for the discrete iterated modified projection solution and use Richardson extrapolation to improve the order of convergence. Our results indicate a choice of a numerical quadrature which preserves the order of convergence in the continuous case.

New fractional signal smoothing equations with short memory and variable order

In this paper, systems of fuzzy fractional differential equations with a lateral type of the Hukuhara derivative and the generalized Hukuhara derivative are numerically studied. Collocation method on discontinuous piecewise polynomial spaces is proposed. Convergence of the proposed method is analyzed. The superconvergent results on the graded mesh are studied. Examples are provided to support theoretical results. Finally, the effect of uncertainty in a diabetes model and its resulting complications is investigated as a practical application.

Richardson Extrapolation of Superconvergent Projection-Type Methods for Hammerstein Equations*

For a nonlinear Hammerstein equation with a smooth kernel, a method proposed recently, based on projection onto a space of piecewise polynomials of degree is shown to have convergence of order This paper shows that this method have asymptotic series expansion and the order of convergence can be further improved to by one step of Richardson extrapolation, assuming the calculation to be repeated with each subinterval halved. Some numerical results are given to illustrate this improvement.

Spline collocation methods for systems of fuzzy fractional differential equations

Abstract In this paper, systems of fuzzy fractional differential equations with a lateral type of the Hukuhara derivative and the generalized Hukuhara derivative are numerically studied. Collocation method on discontinuous piecewise polynomial spaces is proposed. Convergence of the proposed method is analyzed. The superconvergent results on the graded mesh are studied. Examples are provided to support theoretical results. Finally, the effect of uncertainty in a diabetes model and its resulting complications is investigated as a practical application.

Discontinuous Galerkin Difference Methods for Symmetric Hyperbolic Systems

We develop dissipative, energy-stable difference methods for linear first-order hyperbolic systems by applying an upwind, discontinuous Galerkin construction of derivative matrices to a space of discontinuous piecewise polynomials on a structured mesh. The space is spanned by translates of a function spanning multiple cells, yielding a class of implicit difference formulas of arbitrary order. We examine the properties of the method, including the scaling of the derivative operator with method order, and demonstrate its accuracy for problems in one and two space dimensions.

Approximation methods for second kind weakly singular Volterra integral equations

Projection methods are applied to obtain the convergence rates for Volterra integral equations with weakly singular kernels. We consider Galerkin and multi Galerkin methods and their iterated versions to solve Volterra integral equations with weakly singular kernels, in the space of piecewise polynomials subspaces based on graded mesh. We will show that the iterated multi-Galerkin method improves over iterated Galerkin method. In fact, we show that iterated multi-Galerkin solution converges with the convergence rates O(n−3m) and O(n−3m(logn)2), for algebraic and logarithmic type kernels, respectively. We prove that iterated Galerkin method, for algebraic kernel, converges with the convergence rate O(n−2m) and for logarithmic type kernel converges with the convergence rate O(n−2mlogn), where n denotes the number of partition points and m is the highest order of the polynomials employed in the approximations. Theoretical results are justified by the numerical results.

Stable broken $H^1$ and $H(\mathrm {div})$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions

We study extensions of piecewise polynomial data prescribed on faces and possibly in elements of a patch of simplices sharing a vertex. In the $H^1$ setting, we look for functions whose jumps across the faces are prescribed, whereas in the ${\bf H}(\mathrm{div})$ setting, the normal component jumps and the piecewise divergence are prescribed. We show stability in the sense that the minimizers over piecewise polynomial spaces of the same degree as the data are subordinate in the broken energy norm to the minimizers over the whole broken $H^1$ and ${\bf H}(\mathrm{div})$ spaces. Our proofs are constructive and yield constants independent of the polynomial degree. One particular application of these results is in a posteriori error analysis, where the present results justify polynomial-degree-robust efficiency of potential and flux reconstructions.

High-order hybridisable discontinuous Galerkin method for the gas kinetic equation

ABSTRACT The high-order hybridisable discontinuous Galerkin (HDG) method is used to find steady-state solution of gas kinetic equations on two-dimensional geometry. The velocity distribution function and its traces are approximated in piecewise polynomial space on triangular mesh and mesh skeleton, respectively. By employing a numerical flux derived from the upwind scheme and imposing its continuity on mesh skeleton, the global system for unknown traces is obtained with fewer coupled degrees of freedom, compared to the original DG method. The solutions of model equation for the Poiseuille flow through square channel show the higher order solver is faster than the lower order one. Moreover, the HDG scheme is more efficient than the original DG method when the degree of approximating polynomial is larger than 2. Finally, the developed scheme is extended to solve the Boltzmann equation with full collision operator, which can produce accurate results for shear-driven and thermally induced flows.

O3o3: A Variant of Spectral Elements with a Regular Collocation Grid

AbstractIn this study, an alternative local Galerkin method (LGM), the o3o3 scheme, is proposed. o3o3 is a variant or generalization of the third-order spectral element method (SEM3). It uses third-order piecewise polynomials for the representation of a field and piecewise third-degree polynomials for fluxes. For the discretization, SEM3 uses the irregular Legendre–Gauss–Lobatto grid while o3o3 uses a regular collocation grid. o3o3 can be regarded as an inhomogeneous finite-difference scheme on a uniform grid, which means that the finite-difference equations are different for each group with three points. A particular version of o3o3 is set as an example of many possibilities to construct LGM schemes on piecewise polynomial spaces in which the basis functions used are continuous at corner points and function spaces having continuous derivatives are shortly discussed. We propose a standard o3o3 scheme and a spectral o3o3 scheme as alternatives to the standard method of using the quadrature approximation. These two particular schemes selected were chosen for ease of implementation rather than optimal performance. In one dimension, compared to standard SEM3, o3o3 has a larger CFL condition benefiting from the use of a regular collocation grid. While SEM3 uses the irregular Legendre–Gauss–Lobatto collocation grid, o3o3 uses a regular grid. This is considered an advantage for physical parameterizations. The shortest resolved wave is marginally smaller than that with SEM3. In two dimensions, o3o3 is implemented on a sparse grid where only a part of the points on the underlying regular grid are used for forecasting.

Collocation methods for integro-differential algebraic equations with index 1

Abstract The notion of the tractability index based on the $\nu $-smoothing property of a Volterra integral operator is introduced for general systems of linear integro-differential algebraic equations (IDAEs). It is used to decouple the given IDAE system of index $1$ into the inherent system of regular Volterra integro-differential equations (VIDEs) and a system of second-kind Volterra integral equations (VIEs). This decoupling of the given general IDAE forms the basis for the convergence analysis of the two classes of piecewise polynomial collocation methods for solving the given index-$1$ IDAE system. The first one employs the same continuous piecewise polynomial space $S_m^{(0)}$ for both the VIDE part and the second-kind VIE part of the decoupled system. In the second one the VIDE part is discretized in $S_m^{(0)}$, but the second-kind VIE part employs the space of discontinuous piecewise polynomials $S_{m - 1}^{(- 1)}$. The optimal orders of convergence of these collocation methods are derived. For the first method, the collocation solution converges uniformly to the exact solution if and only if the collocation parameters satisfy a certain condition. This condition is no longer necessary for the second method; the collocation solution now converges to the exact solution for any choice of the collocation parameters. Numerical examples illustrate the theoretical results.

Superconvergent product integration method for Hammerstein integral equations

In this paper, we define a superconvergent projection method for approximating the solution of Hammerstein integral equations of the second kind. The projection is chosen either to be the orthogonal or an interpolatory projection at Gauss points onto the space of discontinuous piecewise polynomials. For a smooth kernel or one less smooth along the diagonal, the order of convergence of the proposed method improves upon the classical product integration method. Several numerical examples are given to demonstrate the effectiveness of the current method.

Richardson extrapolation based on superconvergent Nyström and degenerate kernel methods

For computing the approximated solution of a second kind integral equation with a smooth kernel, we investigate in this paper the Richardson extrapolation using superconvergent Nystrom and degenerate kernel methods based on interpolatory projection onto the space of (discontinuous) piecewise polynomials of degree \(\le r - 1.\) We obtain asymptotic series expansions for the approximate solutions and we show that the order of convergence 4r in the interpolation at Gauss points can be improved to \(4r+2\). We illustrate the improvement of the order of convergence by numerical experiments.

Analysis of collocation methods for nonlinear Volterra integral equations of the third kind

We study the approximation of solutions of a class of nonlinear Volterra integral equations (VIEs) of the third kind by using collocation in certain piecewise polynomial spaces. If the underlying Volterra integral operator is not compact, the solvability of the collocation equations is generally guaranteed only if special (so-called modified graded) meshes are employed. It is then shown that for sufficiently regular data the collocation solutions converge to the analytical solution with the same optimal order as for VIEs with compact operators. Numerical examples are given to verify the theoretically predicted orders of convergence.

A Priori Error Estimates for the Optimal Control of the Integral Fractional Laplacian

We design and analyze solution techniques for a linear-quadratic optimal control problem involving the integral fractional Laplacian. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. We propose two strategies to discretize the fractional optimal control problem: a semidiscrete approach where the control is not discretized - the so-called variational discretization approach - and a fully discrete approach where the control variable is discretized with piecewise constant functions. Both schemes rely on the discretization of the state equation with the finite element space of continuous piecewise polynomials of degree one. We derive a priori error estimates for both solution techniques. We illustrate the theory with two-dimensional numerical tests.

The Convergence of Collocation Solutions in Continuous Piecewise Polynomial Spaces for Weakly Singular Volterra Integral Equations

Collocation solutions by globally continuous piecewise polynomials to second-kind Volterra integral equations (VIEs) with smooth kernels are uniformly convergent only for certain sets of collocatio...

Mixed virtual element methods for elastodynamics with weak symmetry

We propose and analyze a mixed virtual element method for linear elastodynamics in velocity–stress formulation with weak symmetry. In this formulation, the symmetry of the stress is relaxed by the rotation of the displacement, and the system of second order differential equation in time is reduced to first order differential equations in time by introducing velocity. The proposed method uses H(div)-conforming virtual element space of order k(k≥1) for the stress and discontinuous piecewise-polynomial spaces of degree k for the velocity and rotation. For time discretization, we use the Crank–Nicolson scheme. Both semidiscrete and fully discrete error estimates are robust for nearly incompressible materials. Numerical experiments confirm our theoretical predictions.

Interface spaces for the Multiscale Robin Coupled Method in reservoir simulation

The Multiscale Robin Coupled Method (MRCM) is a recent multiscale numerical method based on a non-overlapping domain decomposition procedure. One of its hallmarks is that the MRCM allows for the independent definition of interface spaces for pressure and flux over the skeleton of the decomposition. The accuracy of the MRCM depends on the choice of these interface spaces, as well as on an algorithmic parameter β in the Robin interface conditions imposed at the subdomain boundaries. This work presents an extensive numerical assessment of the MRCM in both of these aspects. Two types of interface spaces are implemented: usual piecewise polynomial spaces and informed spaces, the latter obtained from sets of snapshots by dimensionality reduction. Different distributions of the unknowns between pressure and flux are explored. Two non-dimensionalizations of β are tested. The assessment is conducted on realistic, high contrast, channelized permeability fields from a SPE benchmark database. The results show that β, suitably non-dimensionalized, can be fixed to unity to avoid any indeterminacy in the method. Further, with both types of spaces it is observed that a balanced distribution of the interface unknowns between pressure and flux renders the MRCM quite attractive both in accuracy and in computational cost, competitive with other multiscale methods from the literature

Collocation methods for fractional differential equations involving non-singular kernel

A system of fractional differential equations involving non-singular Mittag-Leffler kernel is considered. This system is transformed to a type of weakly singular integral equations in which the weak singular kernel is involved with both the unknown and known functions. The regularity and existence of its solution is studied. The collocation methods on discontinuous piecewise polynomial space are considered. The convergence and superconvergence properties of the introduced methods are derived on graded meshes. Numerical results provided to show that our theoretical convergence bounds are often sharp and the introduced methods are efficient. Some comparisons and applications are discussed.