Local Approximation Order Research Articles

Linearized continuous Galerkin hp‐FEM applied to nonlinear initial value problems

In this note we consider the continuous Galerkin time‐stepping method of arbitrary order as a possible discretization scheme of nonlinear initial value problems. In addition, we develop and generalize a well‐known existing result for the discrete solution by applying a general linearizing procedure to the nonlinear discrete scheme including also the simplified Newton solution procedure. In particular, the presented existence results are implied by choosing sufficient small time steps locally. Furthermore, the established existence results are independent of the local approximation order. Moreover, we will see that the proposed solution scheme is able to significantly reduce the number of iterations. Finally, based on existing and well‐known a priori error estimates for the discrete solution, we present some numerical experiments that highlight the proposed results of this note.

Mollified finite element approximants of arbitrary order and smoothness

The approximation properties of the finite element method can often be substantially improved by choosing smooth high-order basis functions. It is extremely difficult to devise such basis functions for partitions consisting of arbitrarily shaped polytopes. We propose the mollified basis functions of arbitrary order and smoothness for partitions consisting of convex polytopes. On each polytope an independent local polynomial approximant of arbitrary order is assumed. The basis functions are defined as the convolutions of the local approximants with a mollifier. The mollifier is chosen to be smooth, to have a compact support and a unit volume. The approximation properties of the obtained basis functions are governed by the local polynomial approximation order and mollifier smoothness. The convolution integrals are evaluated numerically first by computing the boolean intersection between the mollifier and the polytope and then applying the divergence theorem to reduce the dimension of the integrals. The support of a basis function is given as the Minkowski sum of the respective polytope and the mollifier. The breakpoints of the basis functions, i.e. locations with non-infinite smoothness, are not necessarily aligned with polytope boundaries. Furthermore, the basis functions are not boundary interpolating so that we apply boundary conditions with the non-symmetric Nitsche method as in immersed/embedded finite elements. The presented numerical examples confirm the optimal convergence of the proposed approximation scheme for Poisson and elasticity problems.

An hp-version of the discontinuous Galerkin time-stepping method for Volterra integral equations with weakly singular kernels

We develop and analyze an hp-version of the discontinuous Galerkin time-stepping method for linear Volterra integral equations with weakly singular kernels. We derive a priori error bound in the L2-norm that is fully explicit in the local time steps, the local approximation orders, and the local regularity of the exact solutions. For solutions with singular behavior near t=0 caused by the weakly singular kernels, we prove optimal algebraic convergence rates for the h-version of the discontinuous Galerkin approximations on graded meshes. Moreover, we show that exponential rates of convergence can be achieved for solutions with start-up singularities by using geometrically refined time steps and linearly increasing approximation orders. Numerical experiments are presented to illustrate the theoretical results.

An hp-version of the C0-continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems

We present an hp-version of the C0-continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems. We derive a priori error bound in the H1-norm that is fully explicit with respect to the local time steps and the local approximation orders. Moreover, we prove that the hp-version of the C0-continuous Petrov-Galerkin time stepping method based on geometrically refined time steps and on linearly increasing approximation orders yields exponential rates of convergence for solutions with start-up singularities. Numerical examples confirm the theoretical results.

Equivalent-PDE based stabilization of strong-form meshless methods applied to advection-dominated problems

In this paper a new criterion for stability of strong-form meshless discretizations of advection-dominated problems is developed. The criterion follows from the analysis of the equivalent PDE. It incorporates the influence of the positions of the nodes in the local subdomain, the influence of free parameters of local approximation, such as the weight function and the order of local approximation, and the time stepping formula used. We use the developed criterion in an adaptive upwinding scheme and test the developed scheme on a simple two-dimensional advection problem for regular and irregular node arrangement by using second-, third- and fourth-order accurate explicit diffuse approximate method. The proposed upwinding algorithm significantly improves the accuracy of the resulting solution in comparison with the existing approaches which do not consider the parameters of the local approximation and the irregular positioning of nodes, which is the usual setting for meshless methods.

An hp-version of the C0-continuous Petrov-Galerkin method for second-order Volterra integro-differential equations

We present an hp-version of the C0-continuous Petrov-Galerkin method for linear second-order Volterra integro-differential equations. We combine the continuous and discontinuous Galerkin methodologies to obtain a natural discretization of the second-order derivative of arbitrary order. We derive a-priori error bounds in the L2- and H1-norm that are completely explicit in the local time steps, local approximation orders, and local regularity of the exact solution. Numerical experiments are provided to illustrate the theoretical results.

Second order monotone difference schemes with approximation on non-uniform grids for two-dimensional quasilinear parabolic convection-diffusion equations

The present communication is devoted to the construction of monotone difference schemes of the second order of local approximation on non-uniform grids in space for 2D quasi-linear parabolic convection-diffusion equation. With the help of difference maximum principle, two-sided estimates of the difference solution are established and an important a priori estimate in a uniform norm C is proved. It is interesting to note that the maximal and minimal values of the difference solution do not depend on the diffusion and convection coefficients.

An h–p Version of the Discontinuous Galerkin Method for Volterra Integro-Differential Equations with Vanishing Delays

We present an h–p version of the discontinuous Galerkin time stepping method for Volterra integro-differential equations with vanishing delays. We derive a priori error bounds in the $$L^2$$- and $$L^\infty $$-norm that are explicit in the local time steps, the local approximation orders, and the local regularity of the exact solution. Moreover, we prove that the h–p version of the discontinuous Galerkin scheme based on geometrically refined time steps and on linearly increasing approximation orders achieves exponential rates of convergence for solutions with start-up singularities. Numerical experiments are presented to illustrate the theoretical results.

An [formula omitted]-version of the discontinuous Galerkin time-stepping method for nonlinear second-order delay differential equations with vanishing delays

We present and analyze an hp-version of the discontinuous Galerkin (DG) time-stepping method for nonlinear second-order delay differential equations with vanishing delays. We derive a priori error bounds in the L2- and L∞-norm that are fully explicit with respect to the local time steps, the local approximation orders, and the local regularity of the exact solution. We further prove that the hp-DG scheme based on geometrically refined time steps and on linearly increasing approximation orders attains exponential rates of convergence for analytic solutions with start-up singularities. Moreover, we also propose an hp-DG scheme for the state dependent delay differential equations. We illustrate the theoretical results with a series of numerical experiments.

Direction selection in stochastic directional distance functions

Researchers rely on the distance function to model multiple product production using multiple inputs. A stochastic directional distance function (SDDF) allows for noise in potentially all input and output variables. Yet, when estimated, the direction selected will affect the functional estimates because deviations from the estimated function are minimized in the specified direction. Specifically, the parameters of the parametric SDDF are point identified when the direction is specified; we show that the parameters of the parametric SDDF are set identified when multiple directions are considered. Further, the set of identified parameters can be narrowed via data-driven approaches to restrict the directions considered. We demonstrate a similar narrowing of the identified parameter set for a shape constrained nonparametric method, where the shape constraints impose standard features of a cost function such as monotonicity and convexity.Our Monte Carlo simulation studies reveal significant improvements, as measured by out of sample radial mean squared error, in functional estimates when we use a directional distance function with an appropriately selected direction and the errors are uncorrelated across variables. We show that these benefits increase as the correlation in error terms across variables increase. This correlation is a type of endogeneity that is common in production settings. From our Monte Carlo simulations we conclude that selecting a direction that is approximately orthogonal to the estimated function in the central region of the data gives significantly better estimates relative to the directions commonly used in the literature. For practitioners, our results imply that selecting a direction vector that has non-zero components for all variables that may have measurement error provides a significant improvement in the estimator’s performance. We illustrate these results using cost and production data from samples of approximately 500 US hospitals per year operating in 2007, 2008, and 2009, respectively, and find that the shape constrained nonparametric methods provide a significant increase in flexibility over second order local approximation parametric methods.

Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up

We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to first-order initial value ordinary differential equation problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.

An h-p version of the continuous Petrov-Galerkin method for Volterra delay-integro-differential equations

We consider an h-p version of the continuous Petrov-Galerkin time stepping method for Volterra integro-differential equations with proportional delays. We derive a priori error bounds in the L2-, H1- and L∞-norm that are explicit in the local time steps, the local approximation orders, and the local regularity of the exact solution. Numerical experiments are presented to illustrate the theoretical results.

A high order local approximation free from linear dependency with quadrilateral mesh as mathematical cover and applications to linear elastic fractures

The numerical manifold method falls into the category of the partition of unity methods. In order to enhance accuracy, high order polynomials can be specified as the local approximations. This, however, would incur rank deficiency of the stiffness matrix. In this study, a local displacement approximation is constructed over a physical patch generated from a four quadrilateral mathematical mesh. All the degrees of freedom are physically meaningful. The stresses are continuous at all nodes, suggesting that no stress polish is required. The proposed approximations have the same accuracy as the first-order polynomials, but no linear dependency inherent in the latter.

Transport of phase space densities through tetrahedral meshes using discrete flow mapping

Discrete flow mapping was recently introduced as an efficient ray based method determining wave energy distributions in complex built up structures. Wave energy densities are transported along ray trajectories through polygonal mesh elements using a finite dimensional approximation of a ray transfer operator. In this way the method can be viewed as a smoothed ray tracing method defined over meshed surfaces. Many applications require the resolution of wave energy distributions in three-dimensional domains, such as in room acoustics, underwater acoustics and for electromagnetic cavity problems. In this work we extend discrete flow mapping to three-dimensional domains by propagating wave energy densities through tetrahedral meshes. The geometric simplicity of the tetrahedral mesh elements is utilised to efficiently compute the ray transfer operator using a mixture of analytic and spectrally accurate numerical integration. The important issue of how to choose a suitable basis approximation in phase space whilst maintaining a reasonable computational cost is addressed via low order local approximations on tetrahedral faces in the position coordinate and high order orthogonal polynomial expansions in momentum space.

Stable Simplex Spline Bases for $$C^3$$ C 3 Quintics on the Powell–Sabin 12-Split

For the space of $$C^3$$ quintics on the Powell–Sabin 12-split of a triangle, we determine explicitly the six symmetric simplex spline bases that reduce to a B-spline basis on each edge and have a positive partition of unity, a Marsden identity that splits into real linear factors, and an intuitive domain mesh. The bases are stable in the $$L_\infty $$ norm with a condition number independent of the geometry and have a well-conditioned Lagrange interpolant at the domain points and a quasi-interpolant with local approximation order 6. We show an $$h^2$$ bound for the distance between the control points and the values of a spline at the corresponding domain points. For one of these bases, we derive $$C^0$$ , $$C^1$$ , $$C^2$$ , and $$C^3$$ conditions on the control points of two splines on adjacent macrotriangles.

Simulation of near-field plasmonic interactions with a local approximation order discontinuous Galerkin time-domain method

During the last ten years, the discontinuous Galerkin time-domain (DGTD) method has progressively emerged as a viable alternative to well established finite-difference time-domain (FDTD) and finite-element time-domain (FETD) methods for the numerical simulation of electromagnetic wave propagation problems in the time-domain. The method is now actively studied in various application contexts including those requiring to model light/matter interactions on the nanoscale. Several recent works have demonstrated the viability of the DGDT method for nanophotonics. In this paper we further demonstrate the capabilities of the method for the simulation of near-field plasmonic interactions by considering more particularly the possibility of combining the use of a locally refined conforming tetrahedral mesh with a local adaptation of the approximation order.

Local high-order regularization and applications to hp-methods

We develop a regularization operator based on smoothing on a spatially varying length scale. This operator is defined for functions u∈L1 and has approximation properties that are given by the local Sobolev regularity of u and the local smoothing length scale. Additionally, the regularized function satisfies inverse estimates commensurate with the approximation orders. By combining this operator with a classical hp-interpolation operator, we obtain an hp-Clément type quasi-interpolation operator, i.e., an operator that requires minimal smoothness of the function to be approximated but has the expected approximation properties in terms of the local mesh size and polynomial degree. As a second application, we consider residual error estimates in hp-boundary element methods that are explicit in the local mesh size and the local approximation order.

Effortless quasi-interpolation in hierarchical spaces

We present a general and simple procedure to construct quasi-interpolants in hierarchical spaces. Such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement. The proposed hierarchical quasi-interpolants are described in terms of the so-called truncated hierarchical basis. Assuming a quasi-interpolant is selected for each space associated with a particular level in the hierarchy, the hierarchical quasi-interpolants are obtained without any additional manipulation. The main properties (like polynomial reproduction) of the quasi-interpolants selected at each level are locally preserved in the hierarchical construction. We show how to construct hierarchical local projectors, and the local approximation order of the underling hierarchical space is also investigated. The presentation is detailed for the truncated hierarchical B-spline basis, and we discuss its extension to a more general framework.

Evaluating linear approximations in a two-country model with occasionally binding borrowing constraints

AbstractUnder a linear approximation, a standard two-country business cycles model with incomplete markets delivers consumption and debt dynamics that are non-stationary (unit root) and a bond price that is independent of the wealth distribution. We argue that these two features are due to the local nature of the approximation and we show that they survive even when second or third order local approximations are used. However, these features disappear when debt limits and the associated precautionary motives are taken into account by a standard, global solution method. We subsequently investigate whether this qualitative difference has significant quantitative implications regarding the linear solution as an approximation to the model’s equilibrium dynamics. Policy function differences between the local and global solutions can be large and remain significant even in the case of debt limits as loose as the natural debt limit. These differences can lead to significant discrepancies in implied simulated second moments. In a benchmark calibration, the cross-country correlation of consumption is 0.61 under linearization, but only 0.38 when a policy iteration algorithm is used.

Higher order multipoint method – from Collatz to meshless FDM

The higher order multipoint meshless method for boundary value problems is considered in this paper. The new method relies on the Collatz multipoint concept and the meshless FDM.The main idea of multipoint technique is based on raising the local approximation order of a searched function by assuming additional degrees of freedom at each node of stencil, e.g. by including a combination of nodal values of the right-hand side of considered differential equation. Thus, the FD formula takes into account a combination of unknown function values which is equal a combination of additional d.o.f., e.g. right-hand side of PDE. In this way one may generate higher order FD operators without any additional unknowns using the same set of nodes in stencil as in the non-multipoint case.Essential modifications and extensions of the old classical multipoint formulation have been introduced for the purpose of this research. New fully automatic multipoint meshless FDM uses the moving weighted least squares approximation instead of the interpolation proposed by Collatz, and is based on arbitrarily distributed cloud of nodes. Moreover, besides the local formulation, also various global formulations of b.v. problems are possible.Several numerical benchmark problems analyzed illustrate the effectiveness of the proposed approach.