A finite element method for elasticity systems with discontinuities in the coefficients and the flux across an arbitrary interface is proposed in this paper. The method is based on a Cartesian mesh with local modifications to the mesh. The total degrees of the freedom of the finite element method remains the same as that of the Cartesian mesh. The local modifications lead to a quasi-uniform body-fitted mesh from the original Cartesian mesh. The standard finite element theory and implementation are applicable. Numerical examples that involve discontinuous material coefficients and non-homogeneous jump in the flux across the interface demonstrate the efficiency of the proposed method.

This paper is devoted to the convergence analysis of stochastic $\theta$-methods for nonlinear neutral stochastic differential delay equations (NSDDEs) in Itô sense. The basic idea is to reformulate the original problem eliminating the dependence on the differentiation of the solution in the past values, which leads to a stochastic differential algebraic system. Drift-implicit stochastic $\theta$-methods are proposed for the coupled system. It is shown that the stochastic $\theta$-methods are mean-square convergent with order 1/2 for Lipschitz continuous coefficients of underlying NSDDEs. A nonlinear numerical example illustrates the theoretical results.

The paper deals with convergence and stability of the semi-implicit Euler method with variable stepsize for a linear stochastic pantograph differential equation (SPDE). It is proved that the semi-implicit Euler method with variable stepsize is convergent with strong order $p = \frac{1}{2}$. The conditions under which the method is mean square stability are determined and the numerical experiments are given.

The problem of non-local nonlinear non-Fickian polymer diffusion as modelled by a diffusion equation with a nonlinearly coupled boundary value problem for a viscoelastic 'pseudostress' is considered (see, for example, DA Edwards in Z. angew. Math. Phys., 52, 2001, pp. 254-288). We present two numerical schemes using the implicit Euler method and also the Crank-Nicolson method. Each scheme uses a Galerkin finite element method for the spatial discretisation. Special attention is paid to linearising the discrete equations by extrapolating the value of the nonlinear terms from previous time steps. A priori error estimates are given, based on the usual assumptions that the exact solution possesses certain regularity properties, and numerical experiments are given to support these error estimates. We demonstrate by example that although both schemes converge at their optimal rates the Euler method may be more robust than the Crank-Nicolson method for problems of practical relevance.

In this paper, we analyze a local discontinuous Galerkin method for the willmore flow of graphs. We derive the optimal error estimates for this nonlinear equation in one-dimension and in multi-dimensions for Cartesian meshes using completely discontinuous piecewise polynomial space with degree $k \leq 1$.

This paper is to develop immersed finite element (IFE) functions for solving second order elliptic boundary value problems with discontinuous coefficients and non-homogeneous jump conditions. These IFE functions can be formed on meshes independent of interface. Numerical examples demonstrate that these IFE functions have the usual approximation capability expected from polynomials employed. The related IFE methods based on the Galerkin formulation can be considered as natural extensions of those IFE methods in the literature developed for homogeneous jump conditions, and they can optimally solve the interface problems with a nonhomogeneous flux jump condition.

This paper presents a mixed variational formulation and its discretization by finite elements of higher-order for the Signorini problem with Tresca friction. To guarantee the unique existence of the solution to the discrete mixed problem, a discrete inf-sup condition is proved. Moreover, a solution scheme based on the dual formulation of the problem is proposed. Numerical results confirm the theoretical findings.

In this paper, a semidiscrete finite element Galerkin method for the equations of motion arising in the 2D Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^\infty$ in time, is analyzed. A step-by-step proof of the estimate in the Dirichlet norm for the velocity term which is uniform in time is derived for the nonsmooth initial data. Further, new regularity results are obtained which reflect the behavior of solutions as $t \rightarrow 0$ and $t \rightarrow \infty$. Optimal $L^\infty(L^2)$ error estimates for the velocity which is of order $O(t^{-\frac{1}{2}}h^2)$ and for the pressure term which is of order $O(t^{-\frac{1}{2}}h)$ are proved for the spatial discretization using conforming elements, when the initial data is divergence free and in $H^1_0$ . Moreover, compared to the results available in the literature even for the Navier-Stokes equations, the singular behavior of the pressure estimate as $t \rightarrow 0$, is improved by an order $\frac{1}{2}$, from $t^{-1}$ to $t^{-\frac{1}{2}}$, when conforming elements are used. Finally, under the uniqueness condition, error estimates are shown to be uniform in time.

In this paper, two numerical schemes for finding approximate solutions of singular two-point boundary value problems arising in physiology are presented. While the main ingredient of both approaches is the employment of cubic B-splines, the obstacle of singularity has to be removed first. In the first approach, L'Hopital's rule is used to remove the singularity due to the boundary condition (BC) $y'(0) = 0$. In the second approach, the economized Chebyshev polynomial is implemented in the vicinity of the singular point due to the BC $y(0) = A$, where $A$ is a constant. Numerical examples are presented to demonstrate the applicability and efficiency of the methods on one hand and to confirm the second order convergence on the other hand.