A new adaptive quadrature algorithm that places a greater emphasis on cost reduction while still maintaining an acceptable accuracy is demonstrated. The different needs of science and engineering applications are highlighted as the existing algorithms are shown to be inadequate. The performance of the new algorithm is compared with the well known adaptive Simpson, Gauss-Lobatto and GaussKronrod...

This work presents a study on the performance of nodal bases on triangles and on quadrilaterals for discontinuous Galerkin solutions of hyperbolic conservation laws. A nodal basis on triangles and two tensor product nodal bases on quadrilaterals are considered. The quadrilateral element bases are constructed from the Lagrange interpolating polynomials associated with the Legendre–Gauss–Lobatto ...

Barycentric interpolation is arguably the method of choice for numerical polynomial interpolation. The polynomial interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the interpolation points. Few explicit formulae for these barycentric weights are known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861–877], the authors have shown t...

It is shown that the Kronrod extension to the «-point Gauss integration rule, with respect to the weight function (1 x2)V~"2, 0 < M < 2, ju i= 1, is of exact precision 3n + 1 for n even and 3n + 2 for n odd. Similarly, for the (n-t-l)-point Lobatto rule, with — V¡ < M < 1, u ^ 0, the exact precision is 3n for n odd and 3n + 1 for n even.

We propose and analyze a C spectral element method for a model eigenvalue problem with discontinuous coefficients in the one dimensional setting. A super-geometric rate of convergence is proved for the piecewise constant coefficients case and verified by numerical tests. Furthermore, the asymptotical equivalence between a Gauss-Lobatto collocation method and a spectral Galerkin method is establ...

We develop and analyze a class of maximum bound preserving schemes for approximately solving Allen–Cahn equations. apply kth-order single-step scheme in time (where the nonlinear term is linearized by multi-step extrapolation), lumped mass finite element method space with piecewise rth-order polynomials Gauss–Lobatto quadrature. At each level, cut-off post-processing proposed to eliminate extra...

A singularly perturbed reaction-diiusion equation in two dimensions is considered. We assume analyticity of the input data, i.e., the boundary of the domain is an analytic curve and the right hand side is analytic. We show that the hp version of the nite element method leads to robust exponential convergence provided that one layer of needle elements of width O(p") is inserted near the domain b...

We complete and extend the asymptotic analysis of the spectrum of Jacobi Tau approximations that were first considered by Dubiner. The asymptotic formulas for Jacobi polynomials N P ( , ) , , 1 > − α β α β are derived and confirmed by numerical approximations. More accurate results for the slowest decaying mode are obtained. We explain where the large negative eigenvalues come from. Furthermore...

in the present paper, we develop a modified pseudospectral scheme for solving an optimal control problem which is governed by a switched dynamical system. many real-world processes such as chemical processes, automotive systems and manufacturing processes can be modeled as such systems. for this purpose, we replace the problem with an alternative optimal control problem in which the switching t...