Least‐squares virtual element method for the convection‐diffusion‐reaction problem

Abstract

In this paper, we introduce a least‐squares virtual element method for the convection‐diffusion‐reaction problem in mixed form. We use the H(div) virtual element and continuous virtual element to approximate the flux and the primal variables, respectively. The method allows for the use of very general polygonal meshes. Optimal order a priori error estimates are established for the flux and the primal variables in suitable norms. The least‐squares method offers an efficient a posteriori error estimator without extra effort. Moreover, the hanging nodes are naturally treated in the virtual element method, which provides the high flexibility in mesh refinement because the local mesh postprocessing is never required. Both attractive features motivate us to develop the a posteriori error estimate of the method. Numerical experiments are shown to illustrate the accuracy of the theoretical analysis and demonstrate that the adaptive mesh refinement driven by the proposed estimator can efficiently capture the boundary and the interior layers.

High-Effectiveness and -Accuracy Difference Scheme Based on Nonuniform Grids for Solving Convection–Diffusion Equations with Boundary Layers

“…In this subsection, the RHOC scheme for model Equation ( 5) will be derived based on difference scheme (17). Suppose that model Equation ( 5) has the following difference scheme:…”

An enriched virtual element method for 2D‐3C generalized membrane shell model on surface

“…It doesn't need to define explicit expressions of shape functions and only uses degrees of freedom to calculate the stiffness matrix, which is very convenient and effective. In recent years, this method has been widely applied in various fields, such as Poisson problem, 28,33 elasticity problems, [34][35][36][37][38][39][40][41] plate bending problem, [42][43][44][45][46] Stokes problem, [47][48][49] Navier-Stokes problem, 50,51 contact problem, 52 Steklov eigenvalue problem, 53 Cahn-Hilliard problem, 54 Helmholtz problems, 55 others partial differential equations [56][57][58][59][60][61] and so on. It is worth noting that VEM has yet to be studied in the GMS model.…”