Error analysis of higher order trace finite element methods for the surface Stokes equation

The paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in ℝ3. The method employs parametric Pk-Pk−1 finite element pairs on tetrahedral bulk mesh to discretize the Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming from approximate parametric representation of the surface. Numerical experiments include formal convergence studies and an example of the Kelvin--Helmholtz instability problem on the unit sphere.

A Mixed Discontinuous Galerkin Method for the Helmholtz Equation

In this paper, we introduce and analyze a mixed discontinuous Galerkin method for the Helmholtz equation. The mixed discontinuous Galerkin method is designed by using a discontinuous Pp+1−1−Pp−1 finite element pair for the flux variable and the scattered field with p≥0. We can get optimal order convergence for the flux variable in both Hdiv-like norm and L2 norm and the scattered field in L2 norm numerically. Moreover, we conduct the numerical experiments on the Helmholtz equation with perturbation and the rectangular waveguide, which also demonstrate the good performance of the mixed discontinuous Galerkin method.

Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions

We study a fully discretized finite element approximation to variable-order Caputo and Riemann–Liouville time-fractional diffusion equations (tFDEs) in multiple space dimensions, which model solute transport in heterogeneous porous media and related applications. We prove error estimates for the proposed methods, which are discretized on an equidistant or graded temporal partition predetermined by the behavior of the variable order at the initial time, only under the regularity assumptions of the variable order, coefficients and the source term but without any regularity assumption of the true solutions. Roughly, we prove that the finite element approximations to variable-order Caputo tFDEs have optimal-order convergence rates on a uniform temporal partition. In contrast the finite element approximations to variable-order Riemann–Liouville tFDEs discretized on a uniform temporal partition achieve an optimal-order convergence rate if $\alpha (0)=\alpha ^{\prime}(0) = 0$ but a suboptimal-order convergence rate if $\alpha (0)>0$. In the latter case, optimal-order convergence rate can be proved by employing the graded temporal partition. We conduct numerical experiments to investigate the performance of the numerical methods and to verify the mathematical analysis.

A Framework for Nonconforming Mixed Finite Element Method for Elliptic Problems in ℝ3

In this paper, we suggest a new patch condition for nonconforming mixed finite elements (MFEs) on parallelepiped and provide a framework for the convergence. Also, we introduce a new family of nonconforming MFE space satisfying the new patch condition. The numerical experiments show that the new MFE shows optimal order convergence in Hdiv and L2-norm for various problems with discontinuous coefficient case.

A modified time-fractional diffusion equation and its finite difference method: Regularity and error analysis

The time-fractional diffusion partial differential equations (tFPDEs) (of order 0 < α < 1) properly model the anomalous diffusive transport or memory effects. Recent work [23] showed that the first-order time derivatives of their solutions have a singularity of O(tα−1) near the initial time t = 0, which makes the error estimates of their numerical approximations in the literature that were proved under full regularity assumptions of the true solutions inappropriate. A sharp error estimate was proved for a finite difference method (FDM) with a graded partition for a one-dimensional tFPDE without artificial regularity assumptions on true solutions, [23]. Motivated by the derivation of the tFPDE from stochastic continuous time random walk (CTRW), we present a modified tFPDE and prove that it has full regularity on the entire time interval (including t = 0) and that its FDM on a uniform time partition has an optimal-order convergence rate only under the assumptions of the regularity of the initial condition and right-hand source term. Numerical experiments show that with the same initial data, the solutions of the modified tFPDE and the classical tFPDE converge to each other as time increases, but the solution of the former does not have the singularity as that to the classical tFPDE near time t = 0.

Error Estimates for the Heterogeneous Multiscale Finite Volume Method of Convection-Diffusion-Reaction Problem

Based on the heterogeneous multiscale method, this paper presents a finite volume method to solve multiscale convection-diffusion-reaction problem. The paper constructs an algorithm of the optimal order convergence rate in H1-norm under periodic medias.

A NEW NONCONFORMING MIXED FINITE ELEMENT METHOD FOR LINEAR ELASTICITY

We have developed new nonconforming mixed finite element methods for linear elasticity with a pure traction (displacement) boundary condition based on the Hellinger–Reissner variational principle using rectangular elements. Convergence analysis yields an optimal (suboptimal) convergence rate of [Formula: see text] for the L2-error of the stress and [Formula: see text] for the displacement in the pure traction (displacement) boundary problem. However, numerical experiments have yielded optimal-order convergence rates for both stress and displacement in both problems and have shown superconvergence for the displacement at the midpoint of each element. Moreover, we observed that the optimal convergence rates are still valid for large λ.