In this paper, both semidiscrete and completely discrete finite volume element methods (FVEMs) are analyzed for approximating solutions of a class of linear hyperbolic integro-differential equations in a two-dimensional convex polygonal domain. The effect of numerical quadrature is also examined. In the semidiscrete case, optimal error estimates in $L^∞(L^2)$ and $L^∞(H^1)$ norms are shown to hold with minimal regularity assumptions on the initial data, whereas quasi-optimal estimate is derived in $L^∞(L^∞)$ norm under higher regularity on the data. Based on a second order explicit method in time, a completely discrete scheme is examined and optimal error estimates are established with a mild condition on the space and time discretizing parameters. Finally, some numerical experiments are conducted which confirm the theoretical order of convergence.

In this paper, an explicit a posteriori error estimator is developed for second and fourth order ODEs solved with the Galerkin method that, remarkably, provides exact pointwise error estimates. The error estimator is derived from the variational multiscale theory, in which the subgrid scales are approximated making use of fine-scale Green's functions. This methodology can be extended to any element type and order. Second and fourth order differential equations cover a great variety of problems in mechanics. Two examples with application in elasticity have been studied: the axially loaded beam and the Euler-Bernoulli beam. Because the error estimator is explicit, it can be very easily implemented and its computational cost is very small. Apart from pointwise error estimates, we present local and global a posteriori error estimates in the $L^1$-norm, the $L^2$-norm and the $H^1$-seminorm. Finally, convergence rates of the error and the efficiencies of the estimator are analyzed.

In this paper, we develop and analyze a preconditioning technique and an iterative solver for the linear systems resulting from the discretization of second order elliptic problems by the symmetric interior penalty discontinuous Galerkin methods. The main ingredient of our approach is a stable decomposition of the piecewise polynomial discontinuous finite element space of arbitrary order into a linear conforming space and a space containing high frequency components. To derive such decomposition, we introduce a novel interpolation operator which projects piecewise polynomials of arbitrary order to continuous piecewise linear functions. We prove that this operator is stable which allows us to derive the required space decomposition easily. Moreover, we prove that both the condition number of the preconditioned system and the convergent rate of the iterative method are independent of the mesh size. Numerical experiments are also shown to confirm these theoretical results.

In this paper, we consider an upscaled model describing the multiscale flow of a single-phase incompressible fluid and transport of a dissolved chemical by advection and diffusion through a heterogeneous porous medium. Unlike traditional homogenization or volume averaging techniques, we do not assume a good separation of scales. The new model includes as special cases both the classical homogenized model and the double porosity model, but it is characterized by the presence of additional memory terms which describe the effects of local advective transport as well as diffusion. We study the mathematical properties of the memory (convolution) kernels presented in the model and perform rigorous stability analysis of the numerical method to discretize the upscaled model. Some numerical results will be presented to validate the upscaled model and to show the quantitative significance of each memory term in different regimes of flow and transport.

The finite element dual singular function method [FE-DSFM] has been constructed and analyzed accuracy by Z. Cai and S. Kim to solve the Laplace equation on a polygonal domain with one reentrant corner. In this paper, we impose FE-DSFM to solve the Stokes equations via the mixed finite element method. To do this, we compute the singular and the dual singular functions analytically at a non-convex corner. We prove well-posedness by using the contraction mapping theorem and then estimate errors of the algorithm. We obtain optimal accuracy $O(h)$ for velocity in $\rm{H}^1(Ω)$ and pressure in $L^2(Ω)$, but we are able to prove only $O(h^{1+\lambda})$ error bounds for velocity in $\rm{L}^2(\Omega)$ and stress intensity factor, where $\lambda$ is the eigenvalue (solution of (4)). However, we get optimal accuracy results in numerical experiments.

We study the consistency and convergence of the cell-centered Finite Volume (FV) external approximation of $H^1_0(\Omega)$, where a 2D polygonal domain $\Omega$ is discretized by a mesh of convex quadrilaterals. The discrete FV derivatives are defined by using the so-called Taylor Series Expansion Scheme (TSES). By introducing the Finite Difference (FD) space associated with the FV space, and comparing the FV and FD spaces, we prove the convergence of the FV external approximation by using the consistency and convergence of the FD method. As an application, we construct the discrete FV approximation of some typical elliptic equations, and show the convergence of the discrete FV approximations to the exact solutions.

We first propose an adaptive immersed finite element method based on the a posteriori error estimate for solving elliptic equations with non-homogeneous boundary conditions in general Lipschitz domains. The underlying finite element mesh need not fit the boundary of the domain. Optimal a priori error estimate of the proposed immersed finite element method is proved. The immersed finite element method is then used to solve parabolic problems in time variable domains together with an arbitrary Lagrangian-Eulerian (ALE) time discretization scheme. An a posteriori error estimate for the fully discrete immersed finite element method is derived which can be used to adaptively update the time step sizes and finite element meshes at each time step. Numerical experiments are reported to support the theoretical results.