Abstract A finite element method of any order is applied on a Bakhvalov‐type mesh to solve a singularly perturbed convection–diffusion equation in 2D, whose solution exhibits exponential boundary layers. A uniform convergence of (almost) optimal order is proved by means of a carefully defined interpolant.

Abstract In this article, we study anisotropic immersed finite element (AIFE) methods for the elliptic interface system. After establishing the existence and uniqueness of the weak solution, we construct a linear AIFE method for solving the interface system. Different from the scalar problem, the jump conditions for each unknown function are coupled together in the anisotropic problem considered here. Through defining suitable interpolations, we deduce the error estimates in and for the proposed AIFE method. Moreover, we show some numerical experiments to verify the correction of the theoretical predictions.

Abstract This paper analyzes a class of two‐dimensional (2‐D) time fractional reaction‐subdiffusion equations with variable coefficients. The high‐order L2‐1σ time‐stepping scheme on graded meshes is presented to deal with the weak singularity at the initial time t = 0, and the bilinear finite element method (FEM) on anisotropic meshes is used for spatial discretization. Using the modified discrete fractional Grönwall inequality, and combining the interpolation operator and the projection operator, the L2‐norm error estimation and H1‐norm superclose results are rigorously proved. The superconvergence result in the H1‐norm is derived by applying the interpolation postprocessing technique. Finally, numerical examples are presented to verify the validation of our theoretical analysis.

Abstract Time efficiency is one of the more critical concerns in computational fluid dynamics simulations of industrial applications. Extensive research has been conducted to improve the underlying numerical schemes to achieve time process reduction. Within this context, this paper presents a new time discretization method based on the Adomian decomposition technique for Euler equations. The obtained scheme is time‐order adaptive; the order is automatically adjusted at each time step and over the space domain, leading to significant processing time reduction. The scheme is formulated in an appropriate recursive formula, and its efficiency is demonstrated through numerical tests by comparison to exact solutions and the popular Runge–Kutta‐discontinuous Galerkin method.

Abstract In this article, we introduce and analyze arbitrary‐order, locally conservative hybrid discontinuous Galkerin methods for linearized Navier–Stokes equations. The unknowns of the global system are reduced to trace variables on the skeleton of a triangulation and the average of pressure on each cell via embedded static condensation. We prove that the lifting operator associated with trace variables is injective for any polynomial degree. This generalizes the result in (Y. Jeon and E.‐J. Park, Numerische Mathematik 123 [2013], no. 1, pp. 97–119), where quadratic and cubic rectangular elements are analyzed. Moreover, optimal error estimates in the energy norm are obtained by introducing nonstandard projection operators for the hybrid DG method. Several numerical results are presented to show the performance of the algorithm and to validate the theory developed in the article.

Abstract This research work presents a newly truly meshless approach based upon the integrated radial basis functions (IRBFs) technique to study the fractional modified anomalous sub‐diffusion equation. First, the temporal direction is discretized by a finite difference method with second‐order accuracy. The stability analysis and convergence of the proposed time‐discrete formulation are investigated, theoretically. Then, the spatial direction is approximated by the IRBFs methodology. In this approach, the largest‐order derivative is approximated by a linear combination of RBFs and then the lower‐order derivative and also the unknown function are constructed by repeated integration. According to this procedure, the existing derivatives in the main fractional PDE are approximated smoothly. Furthermore, to show the efficiency of the developed numerical procedure, we employed some non‐rectangular computational domains to obtain the numerical results. On the other hand, the mentioned model is solved in one‐ two‐, and three dimensional cases. The numerical results confirm the ability and efficiency of the new numerical method for solving time fractional PDEs on complex computational domains.

Abstract In this article, we present a parameters‐uniform numerical method for a time‐dependent weakly coupled system of two convection‐diffusion equations that has a discontinuity, along the line in the source term. The second order term of each equation is multiplied by a small singular perturbation parameter and these parameters are assumed to be different in magnitude. On an appropriate Shishkin mesh, the considered problem is discretized using an upwind central difference scheme away from the line of discontinuity, and a particular upwind central difference scheme along the line of discontinuity. The numerical approximations produced by this scheme are uniformly convergent of first‐order in time and almost first‐order in space with respect to both perturbation parameters. Numerical results are presented to validate the theoretical results.