We consider a linear singularly perturbed reaction-diffusion problem in three dimensions and its numerical solution by a Galerkin finite element method with trilinear elements. The problem is discretised on a Shishkin mesh with $N$ intervals in each coordinate direction. Derivation of an error estimate for such a method is usually based on the (Shishkin) decomposition of the solution into distinct layer components. Our contribution is to provide a careful and detailed analysis of the trilinear interpolants of these components. From this analysis it is shown that, in the usual energy norm the errors converge at a rate of $\mathcal{O}$($N$⁻²+$ε$^1/2$N$⁻¹ln$N$). This is validated by numerical results.

This paper proposes a new constrained total generalized variation (TGV)-shearlet model to the compressive sensing magnetic resonance imaging (MRI) reconstruction via the simple parameter estimation scheme. Due to the non-smooth term included in the proposed model, we employ the alternating direction method of multipliers (ADMM) scheme to split the original problem into some easily solvable subproblems in order to use the convenient soft thresholding operator and the fast Fourier transformation (FFT). Since the proposed numerical algorithm belongs to the framework of the classic ADMM, the convergence can be kept. Experimental results demonstrate that the proposed method outperforms the state-of-the-art unconstrained reconstruction methods in removing artifacts and achieves lower reconstruction errors on the tested dataset.

Stability is an important aspect of numerical methods for hyperbolic conservation laws and has received much interest. However, continuity in time is often assumed and only semidiscrete stability is studied. Thus, it is interesting to investigate the influence of explicit and implicit time integration methods on the stability of numerical schemes. If an explicit time integration method is applied, spacially stable numerical schemes for hyperbolic conservation laws can result in unstable fully discrete schemes. Focusing on the explicit Euler method (and convex combinations thereof), undesired terms in the energy balance trigger this phenomenon and introduce an erroneous growth of the energy over time. In this work, we study the influence of artificial dissipation and modal filtering in the context of discontinuous spectral element methods to remedy these issues. In particular, lower bounds on the strength of both artificial dissipation and modal filtering operators are given and an adaptive procedure to conserve the (discrete) L₂ norm of the numerical solution in time is derived. This might be beneficial in regions where the solution is smooth and for long time simulations. Moreover, this approach is used to study the connections between explicit and implicit time integration methods and the associated energy production. By adjusting the adaptive procedure, we demonstrate that filtering in explicit time integration methods is able to mimic the dissipative behavior inherent in implicit time integration methods. This contribution leads to a better understanding of existing algorithms and numerical techniques, in particular the application of artificial dissipation as well as modal filtering in the context of numerical methods for hyperbolic conservation laws together with the selection of explicit or implicit time integration methods.

The paper focuses on numerical study of the incompressible miscible flow in porous media. The proposed algorithm is based on a fully decoupled and linearized scheme in the temporal direction, classical Galerkin-mixed approximations in the FE space ($V^r_h$, $S^{r-1}_h$ × $H^{r-1}_h$) ($r$ ≥ 1) in the spatial direction and a post-processing technique for the velocity/pressure, where $V^r_h$ and $S^{r-1}_h$ × $H^{r-1}_h$ denotes the standard $C^0$ Lagrange FE and the Raviart-Thomas FE spaces, respectively. The decoupled and linearized Galerkin-mixed FEM enjoys many advantages over existing methods. At each time step, the method only requires solving two linear systems for the concentration and velocity/pressure. Analysis in our recent work [37] shows that the classical Galerkin-mixed method provides the optimal accuracy $O$($h^{r+1}$) for the numerical concentration in $L^2$-norm, instead of $O$($h^r$) as shown in previous analysis. A new numerical velocity/pressure of the same order accuracy as the concentration can be obtained by the post-processing in the proposed algorithm. Extensive numerical experiments in both two- and three-dimensional spaces, including smooth and non-smooth problems, are presented to illustrate the accuracy and stability of the algorithm. Our numerical results show that the one-order lower approximation to the velocity/pressure does not influence the accuracy of the numerical concentration, which is more important in applications.

In this paper, we present a finite difference method for stochastic nonlinear second-order boundary-value problems (BVPs) driven by additive noises. We first approximate the white noise process with its piecewise constant approximation to obtain an approximate stochastic BVP. The solution to the new BVP is shown to converge to the solution of the original BVP at $\mathcal{O}$($h$) in the mean-square sense. The approximate BVP is shown to have certain regularity properties which are not true in general for the solution to the original stochastic BVP. The standard finite difference method for deterministic BVPs is then applied to approximate the solution of the new stochastic BVP. Convergence analysis is presented for the numerical solution based on the standard finite difference method. We prove that the finite difference solution converges to the solution to the original stochastic BVP at $\mathcal{O}$($h$) in the mean-square sense. Finally, we perform several numerical examples to validate the theoretical results.

Polynomial Preserving Recovery (PPR) is a popular post-processing technique for finite element methods. In this article, we propose and analyze an effective linear element PPR on the equilateral triangular mesh. With the help of the discrete Green's function, we prove that, when using PPR to the linear element on a specially designed hexagon patch, the recovered gradient can reach $O$($h$⁴| ln $h$|^{$\frac{1}{2}$}) superconvergence rate for the two dimensional Poisson equation. In addition, we apply PPR to the quadratic element on uniform triangulation of the Chevron pattern with an application to the wave equation, which further verifies the superconvergence theory.

The method of numerical simulation based on the splitting by physical processes of gas-hydrodynamic processes, which occur during the dissociation of gas hydrates in a porous medium, is described. In this paper, a coupled discrete model of a two-component ($H$₂$O$, $CH$₄) three-phase (water, methane, hydrate) filtration fluid dynamics and two-phase processes in a thawed zone with absence of gas hydrates in thermodynamic equilibrium has been developed, by using the splitting by physical processes as a valid assumption. The obtained split model is differentially equivalent to the discrete initial balance equations of the system (conservation of the mass components of the fluids and the total energy of the system), written in divergent form. Such an approach to create completely conservative difference schemes in the studied fluid-hydrate medium requires the introduction of a special free-volume nonlinear approximation of grid functions over time, which depends on the volume fraction in the pores occupied by fluids, and is simple to implement. The direct unsplit use of the studied system for the purposes of determining the dynamics of variables and constructing the implicit difference scheme required for calculations of filtering processes with large time steps is difficult. The paper also presents the method of coupled solutions of systems of equations describing the processes in various fields, each of which is characterized by its own set of coexisting phases, and the coordination of computational schemes for them is not an automatic process. In the results of the calculations, the volumetric three-phase phase transitions were numerically investigated using a single calculation with a variable number of phases region of the entire plane of the P and T parameters. Using the example of the Messoyakha's gas hydrate deposit, the local processes of technogenic depressive impact directly near the wells on the dynamics of the gas distribution of gas hydrates thawing and formation of thawed two-phase zones were studied.

An adaptive E–scheme for degenerate, viscous balance laws is presented. Taking into account natural diffusion, numerical viscosity is locally reduced to a minimum. Numerical experiments demonstrate the improved accuracy of the adaptive scheme. Explicit and implicit three–point E–schemes are monotone, TVD and nonlinearly stable. A high–resolution version of the adaptive E–scheme is derived and tested in experiments. The latter is not necessarily monotone, but TVD.