In this work, we present a cell-centered time-splitting technique for solving evolutionary diffusion equations on triangular grids. To this end, we consider three variables (namely the pressure, the flux and a weighted gradient) and construct a so-called expanded mixed finite element method. This method introduces a suitable quadrature rule which permits to eliminate both fluxes and gradients, thus yielding a cell-centered semidiscrete scheme for the pressure with a local 10-point stencil. As for the time integration, we use a domain decomposition operator splitting based on a partition of unity function. Combining this splitting with a multiterm fractional step formula, we obtain a collection of uncoupled subdomain problems that can be efficiently solved in parallel. A priori error estimates for both the semidiscrete and fully discrete schemes are derived on smooth triangular meshes with six triangles per internal vertex.

In this paper we propose a unified formulation to introduce and analyze (pure) Lagrangian and semi-Lagrangian methods for solving convection-diffusion partial differential equations. This formulation allows us to state classical and new numerical methods. Several examples are given. We combine them with finite element methods for spatial discretization. One of the pure Lagrangian methods we introduce has been analyzed in [4] and [5] where stability and error estimates for time semi-discretized and fully-discretized schemes have been proved. In this paper, we prove new stability estimates. More precisely, we obtain an $l^∞(H^1)$ stability estimate independent of the diffusion coefficient and, if the underlying flow is incompressible, we get a stability inequality independent of the final time. Finally, the numerical solution of a test problem is presented that confirms the new stability results.

We present and analyze a subgrid viscosity Lagrange-Galerkin method that combines the subgrid eddy viscosity method proposed in W. Layton, A connection between subgrid scale eddy viscosity and mixed methods. Appl. Math. Comp., 133: 147-157, 2002, and a conventional Lagrange-Galerkin method in the framework of $P_1\oplus$ cubic bubble finite elements. This results in an efficient and easy to implement stabilized method for convection dominated convection-diffusion-reaction problems. Numerical experiments support the numerical analysis results and show that the new method is more accurate than the conventional Lagrange-Galerkin one.

In the present paper a general technique is developed for construction of compact high-order finite difference schemes to approximate Schrödinger problems on nonuniform meshes. Conservation of the finite difference schemes is investigated. The same technique is applied to construct compact high-order approximations of the Robin and Szeftel type boundary conditions. Results of computational experiments are presented.

We consider very weak solutions of a nonlinear version (non-Hookean materials) of the beam stationary Bernoulli-Euler equation, as well as the similar extension to plates, involving the bi-Laplacian operator, with Navier (hinged) boundary conditions. We are specially interested in the case in which the usual Sobolev space framework cannot be applied due to the singularity of the load density near the boundary. We present some properties of such solutions as well as some numerical experiences illustrating how the behaviour of the very weak solutions near the boundary is quite different to the one of more regular solutions corresponding to non-singular load functions.

In this paper, we present a fully discretized Calderόn Calculus for the two dimensional Helmholtz equation. This full discretization can be understood as highly non-conforming Petrov-Galerkin methods, based on two staggered grids of mesh size $h$, Dirac delta distributions substituting acoustic charge densities and piecewise constant functions for approximating acoustic dipole densities. The resulting numerical schemes from this calculus are all of order $h^2$ provided that the continuous equations are well posed. We finish by presenting some numerical experiments illustrating the performance of this discrete calculus.

In many applied problems, the individual components of the unknown vector are interconnected and therefore splitting schemes are applied in order to get a simple problem for evaluating unknowns at a new time level. On the basis of additive schemes (splitting schemes), there are constructed efficient computational algorithms for numerical solving the initial value problems for systems of time-dependent PDEs. The present paper deals with computational algorithms that are based on using explicit-implicit approximations in time. Typically, additive operator-difference schemes for systems of evolutionary equations are constructed for operators that are coupled in space. Here we investigate more general problems, where we have coupling of derivatives in time for components of the solution vector.

A linear singularly perturbed time dependent convection-diffusion problem is examined. The initial condition is designed to have steep gradients in the vicinity of the inflow point, which are transported in time, thus creating a moving interior shock layer. The location of this interior layer is tracked by the characteristics of the reduced first order problem. A numerical method is designed and analysed, which consists of a monotone finite difference operator and a piecewise-uniform Shishkin mesh, which is aligned to the characteristic curve emanating from the initial shock location. Parameter explicit error bounds are established and numerical results are presented to illustrate the performance of the numerical method.

In 1995 the genesis of stabilized methods was established by Professor Hughes from the standpoint of the variational multiscale theory (VMS). By splitting the solution into resolved and unresolved scales, it was unveiled that stabilized methods take into account an approximation of the unresolved scales or error into the finite element solution. In this work, the VMS theory is exploited to formulate an explicit a-posteriori error estimator, consistent with the assumptions inherent to stabilized methods.The proposed technology, which is especially suited for fluid flow problems, is very economical and can be implemented in standard finite element codes. It has been shown that, in practice, the method is robust uniformly from the diffusive to the hyperbolic limit.The success of the method can be explained by the fact that in stabilized methods the element local problems for the fine-scale Green's function capture most of the error and the error intrinsic time-scales are an approximation to the solution of the dual problem. Applications to the Euler and linear elasticity equations are shown.

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with discontinuous source terms is considered. A small positive parameter multiplies the leading term of each equation. These singular perturbation parameters are assumed to be distinct. The components of the solution exhibit overlapping boundary and interior layers. A numerical method is constructed that uses a classical finite difference scheme on a piecewise uniform Shishkin mesh. It is proved that the numerical approximations obtained by this method are essentially first order convergent uniformly with respect to all of the perturbation parameters. Numerical illustrations are presented in support of the theory.

The main purpose of this work is the efficient implementation of a multigrid algorithm for solving Navier-Stokes problems at low Reynolds numbers in different triangular geometries. In particular, a finite element formulation of the Navier-Stokes equations, using quadratic finite elements for the velocities and linear finite elements to approximate the pressure, is used to solve the problem of flow in a triangular cavity, driven by the uniform motion of one of its side walls. An appropriate multigrid method for this discretization of Navier-Stokes equations is designed, based on a Vanka type smoother. Moreover, the data structure used allows an efficient stencil-based implementation of the method, which permits us to perform simulations with a large number of unknowns with low memory consumption and a relatively low computational cost.

A new technique to study special difference schemes numerically for a Dirichlet problem on a rectangular domain (in $x$, $t$) is considered for a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter $\varepsilon$; $\varepsilon\in(0, 1]$. A well known difference scheme on a piecewise-uniform grid is used to solve the problem. Such a scheme converges $\varepsilon$-uniformly in the maximum norm at the rate $\mathcal{O}(N^{-2} ln^2 N + N^{-1}_0)$ as $N$, $N_0 \rightarrow ∞$, where $N+1$ and $N_0+1$  are the numbers of nodes in the spatial and time meshes, respectively; for $\varepsilon\geq m ln^{-1} N$ the scheme converges at the rate $\mathcal{O}(N^{-2} + N^{-1}_0)$. In this paper we elaborate a new approach based on the consideration of regularized errors in discrete solutions, i.e., total errors (with respect to both variables $x$ and $t$), and also fractional errors (in $x$ and in $t$) generated in the approximation of differential derivatives by grid derivatives. The regularized total errors agree well with known theoretical estimates for actual errors and their convergence rate orders. It is also shown that a "standard" approach based on the "fine grid technique" turns out inefficient for numerical study of difference schemes because this technique brings to large errors already when estimating the total actual error.