For the evaluation of free-form polynomial and trigonometric curves and surfaces, several Taylor methods and two new methods (DP and DT) motivated by Computer Aided Geometric Design (CAGD) have been considered. Their accuracy and computational costs are compared through numerical examples. In the polynomial case they are also compared with the most used evaluation algorithms.

In this paper, we post-process an eight-node-serendipity finite element solution for elliptic equations. In the post-processing procedure, we first construct a $control$ $volume$ for each node in the serendipity finite element mesh, then we enlarge the serendipity finite element space by adding some appropriate element-wise bubbles and require the novel solution to satisfy the local conservation law on each control volume. Our post-processing procedure can be implemented in a parallel computing environment and its computational cost is proportional to the cardinality of the serendipity elements. Moreover, both our theoretical analysis and numerical examples show that the postprocessed solution converges to the exact solution with optimal convergence rates both under $H^1$ and $L^2$ norms. A numerical experiment for a single-phase porous media problem validates the necessity of the post-processing procedure.

We consider a class of nonstandard high-order PDEs models, based on the ($p(·)$, $q(·)$)-Kirchhoff operator with variable exponents for the image denoising problem. We theoretically analyse the proposed non-linear model. Then, we use linearization method based on a fixed-point iterative technique and we also prove the convergence of the iterative process. The model has a multiscale character which follows from an adaptive selection of the exponents $p(·)$ and $q(·)$. The latter task helps to capture, highlight and correlate major features in the images and optimize the smoothing effect. We use Morley finite-elements for the numerical resolution of the proposed model and we give several numerical examples and comparisons with different methods.

In this paper, we consider weakly regular Sturm-Liouville eigenproblems with unbounded potential at both endpoints of the domain. We propose a Galerkin spectral matrix method for its solution and we study the error in the eigenvalue approximations it provides. The result of the convergence analysis is then used to derive a low-cost and very effective formula for the computation of corrected numerical eigenvalues. Finally, we present and discuss the results of several numerical experiments which confirm the validity of the approach.

In the paper, an even-odd cycled high-order splitting finite difference time domain scheme for Maxwell's equations in two dimensions is developed. The scheme uses fourth order spatial difference operators and even-odd time step technique to make it more accurate in both space and time. The scheme is energy-conserved, unconditionally stable and efficient in computation. We analyze in detail the stability, dispersion and phase error for the scheme. We prove that the scheme is energy conservative. Numerical experiments show numerically the energy conservation, high accuracy, and the divergence free accuracy. Furthermore, the developed scheme is applied to compute of the grounded coplanar waveguides.

In this paper, a finite difference scheme is established for solving the fourth-order time multi-term fractional sub-diffusion equations with the first Dirichlet boundary conditions. Using the method of order reduction, the original problem is equivalent to a lower-order system. Then the system is considered at some particular points, and the first Dirichlet boundary conditions are also specially handled, so that the global convergence of the presented difference scheme reaches $O(τ^2 + h^4)$, with $τ$ and $h$ the temporal and spatial step size, respectively. The energy method is used to give the theoretical analysis on the stability and convergence of the difference scheme, where some novel techniques have been applied due to the non-local property of fractional operators and the numerical treatment of the first Dirichlet boundary conditions. Numerical experiments further validate the theoretical results.

In this article, we develop a Taylor-Hood immersed finite element (IFE) method to solve two-dimensional Stokes interface problems. The $P_2$-$P_1$ local IFE spaces are constructed using the least-squares approximation on an enlarged fictitious element. The partially penalized IFE method with ghost penalty is employed for solving Stoke interface problems. Penalty terms are imposed on both interface edges and the actual interface curves. Ghost penalty terms are enforced to enhance the stability of the numerical scheme, especially for the pressure approximation. Optimal convergences are observed in various numerical experiments with different interface shapes and coefficient configurations. The effects of the ghost penalty and the fictitious element are also examined through numerical experiments.