In this work, we study a strain gradient problem involving mixtures. The variational formulation is written as a first-order in time coupled system of parabolic variational equations. An existence and uniqueness result is recalled. Then, we introduce a fully discrete approximation by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved. Finally, some one- and two-dimensional numerical simulations are performed.

A solenoidal basis is constructed to compute velocities using a certain finite element method for the Stokes problem. The method is conforming, with piecewise linear velocity and piecewise constant pressure on the Powell-Sabin split of a triangulation. Inhomogeneous Dirichlet conditions are supported by constructing an interpolating operator into the solenoidal velocity space. The solenoidal basis reduces the problem size and eliminates the pressure variable from the linear system for the velocity. A basis of the pressure space is also constructed that can be used to compute the pressure after the velocity, if it is desired to compute the pressure. All basis functions have local support and lead to sparse linear systems. The basis construction is confirmed through rigorous analysis. Velocity and pressure system matrices are both symmetric, positive definite, which can be exploited to solve their corresponding linear systems. Significant efficiency gains over the usual saddle-point formulation are demonstrated computationally.

This paper is devoted to a discontinuous Galerkin (DG) method for nonlinear quasi-static poroelasticity problems. The fully implicit nonlinear numerical scheme is constructed by utilizing DG method for the spatial approximation and the backward Euler method for the temporal discretization. The existence and uniqueness of the numerical solution is proved. Then we derive the optimal convergence order estimates in a discrete $H^1$ norm for the displacement and in $H^1$ and $L^2$ norms for the pressure. Finally, numerical experiments are supplied to validate the theoretical error estimates of our proposed method.

We consider a semi-discrete finite element method for a dynamic model for linear viscoelastic materials based on the constitutive law of fractional order. The corresponding integro-differential equation is of a Mittag-Leffler type convolution kernel. A 4-node hybrid stress quadrilateral finite element is used for the spatial discretization. We show the existence and uniqueness of the semi-discrete solution, then derive some error estimates. Finally, we provide several numerical examples to verify the theoretical results.

In this paper, by introducing a new flux variable, two decoupled and linearized block-centered finite difference methods are developed and analyzed for the nonlinear symmetric regularized long wave equation, where the two-step backward difference formula and Crank-Nicolson temporal discretization combined with linear extrapolation technique are employed. Under a reasonable time stepsize ratio restriction, i.e., $∆t=o(h^{1/4}),$ second-order convergence for both the primal variable and its flux are rigorously proved on general non-uniform spatial grids. Moreover, based upon the convergence results and inverse estimate, stability of two methods are also demonstrated. Ample numerical experiments are presented to confirm the theoretical analysis.

In this work, we consider Richardson extrapolation of the Crank-Nicolson (CN) scheme for backward stochastic differential equations (BSDEs). First, applying the Adomian decomposition to the nonlinear generator of BSDEs, we introduce a new system of BSDEs. Then we theoretically prove that the solution of the CN scheme for BSDEs admits an asymptotic expansion with its coefficients the solutions of the new system of BSDEs. Based on the expansion, we propose Richardson extrapolation algorithms for solving BSDEs. Finally, some numerical tests are carried out to verify our theoretical conclusions and to show the stability, efficiency and high accuracy of the algorithms.

An efficient iteration method is provided in this paper for solving a class of nonlinear systems whose Jacobian matrices are complex and symmetric. The modified Newton-NDSS method is developed and applied to the class of nonlinear systems by adopting the modified Newton method as the outer solver and a new double-step splitting (NDSS) iteration scheme as the inner solver. Additionally, we theoretically analyze the local convergent properties of the new method under the weaker Hölder conditions. Lastly, the new method is compared numerically with some existing ones and the numerical experiments solving the nonlinear equations demonstrate the superiority of the Newton-NDSS method.