In this paper, we develop and analyze an implicit $a$ $posteriori$ error estimates for the local discontinuous Galerkin (LDG) method for nonlinear third-order Korteweg-de Vries (KdV) equations in one space dimension. First, we show that the LDG error on each element can be split into two parts. The first part is proportional to the $(p+1)$-degree right Radau polynomial and the second part converges with order $p$ $+$ $\frac{3}{2}$ in the $L^2$-norm, when piecewise polynomials of degree at most $p$ are used. These results allow us to construct $a$ $posteriori$ LDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these $a$ $posteriori$ error estimates converge at a fixed time to the exact spatial errors in the $L^2$-norm under mesh refinement. The order of convergence is proved to be $p$ $+$ $\frac{3}{2}$. Finally, we prove that the global effectivity index converges to unity at $\mathcal{O}(h^{\frac{1}{2}})$ rate. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed error estimator.

The conforming discontinuous Galerkin (CDG) finite element methods were introduced in [12] on simplicial meshes and in [13] on polytopal meshes. The CDG method gets its name by combining the features of both conforming finite element method and discontinuous Galerkin (DG) finite element method. The goal of this paper is to continue our efforts on simplifying formulations for the finite element method with discontinuous approximation by constructing new spaces for the gradient approximation. Error estimates of optimal order are established for the corresponding CDG finite element approximation in both a discrete $H^1$ norm and the $L^2$ norm. Numerical results are presented to confirm the theory.

The paper is devoted to the construction and study of the decomposition type semi-discrete scheme for one nonlinear multi-dimensional integro-differential equation of parabolic type. Unique solvability of the first type initial-boundary value problem is given as well. The studied equation is some generalization of integro-differential model, which is based on the well-known Maxwell system arising in mathematical simulation of electromagnetic field penetration into a medium.

In this paper, we consider a mathematical model which describes the quasistatic frictionless contact between a viscoplastic body and a foundation. The contact is modeled with normal compliance and unilateral constraint. We present the variational-hemivariational formulation of the model and prove its unique solvability. Then we introduce a fully discrete scheme to solve the problem and derive an error estimate. Under appropriate regularity assumptions of the exact solution, we obtain the optimal order error estimate. Finally, numerical results are reported to show the performance of the numerical method.

In this paper, we use recently developed theories of divergence–free finite element schemes to analyze methods for the Stokes problem with grad-div stabilization. For example, we show that, if the polynomial degree is sufficiently large, the solutions of the Taylor–Hood finite element scheme converges to an optimal convergence exactly divergence–free solution as the grad-div parameter tends to infinity. In addition, we introduce and analyze a stable first-order scheme that does not exhibit locking phenomenon for large grad-div parameters.

A fast algorithm for the nonlocal unsteady problems was proposed, which can be employed in the numerical simulation of nonlocal diffusion and peridynamic. The surrogate model constructed by the proper orthogonal decomposition (POD) speeds up the process of solving equations by reducing the order of linear equations. Then, the accuracy and efficiency of the proposed algorithm was verified by numerical experiments. The results showed that this approach ensures accuracy while reduces the computational burden of the nonlocal model.

In this work, we propose and analyze a class of bubble enriched quadratic finite volume element schemes for anisotropic diffusion problems on triangular meshes. The trial function space is defined as quadratic finite element space by adding a space which consists of element-wise bubble functions, and the test function space is the piecewise constant space. For the class of schemes, under the coercivity result, we proved that $|u − u_h|_1$ = $\mathcal{O}(h^2)$ and $‖u − u_h‖_0$ = $\mathcal{O}(h^3)$, where $u$ is the exact solution and $u_h$ is the bubble enriched quadratic finite volume element solution. The theoretical findings are validated by some numerical examples.

We present corner singularity expansion for the solutions of the Stokes equations in an arbitrary polygon under the stress boundary condition, with explicit expression of its singular part and quantitative estimate of its regular part. In particular, there is a countably infinite set of angles such that a corner with one of these angles would give the solution precisely one additional logarithmic singularity, whose explicit expression is found.