We present and analyze a hybridizable discontinuous Galerkin (HDG) finite element method for the coupled Stokes–Biot problem. Of particular interest is that discrete velocities displacement are H(div)-conforming satisfy compressibility equations pointwise on elements. Furthermore, in incompressible limit, discretization strongly conservative. prove well-posedness of and, after combining HDG wit...

The fractional Fokker-Planck equation is often used to characterize anomalous diffusion. In this paper, a fully discrete approximation for the nonlinear spatial fractional Fokker-Planck equation is given, where the discontinuous Galerkin finite element approach is utilized in time domain and the Galerkin finite element approach is utilized in spatial domain. The priori error estimate is derived...

A novel nonlinear formulation of finite element and Galerkin methods is presented here, which leads to the Hadamard product expression of the resultant nonlinear algebraic analogue. The presented formulation attains the advantages of weak formulation in the standard finite element and Galerkin schemes and avoids the costly repeated numerical integration of the Jacobian matrix via the recently d...

We present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem. Relying on the equilibration by the mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with ...

A simulation of a cantilever beam was carried out with the aim of comparing the performance of two mesh-free methods: element-free Galerkin and radial point interpolation methods. In this implementation, we use two different methods to set boundary conditions. The results were compared with the analytical solution of the cantilever problem using Timoshenko formulation [Tim70]. In the paper, we ...

A numerical method for the simulation of three-dimensional incompressible twophase flows is presented. The proposed algorithm combines an implicit pressure stabilized finite element method for the solution of incompressible two-phase flow problems with a level set method implemented with a quadrature-free Discontinuous Galerkin (DG) method [1]. The use of a fast contouring algorithm [2] permits...

A discontinuous Galerkin finite element method has been developed to treat the high-order spatial derivatives appearing in the Cahn–Hilliard equation. The Cahn–Hilliard equation is a fourth-order nonlinear parabolic partial differential equation, originally proposed to model phase segregation of binary alloys. The developed discontinuous Galerkin approach avoids the need for mixed finite elemen...

This paper presents a new numerical method for the compressible Navier-Stokes equations governing the flow of an ideal isentropic gas. To approximate the continuity equation, the method utilizes a discontinuous Galerkin discretization on piecewise constants and a basic upwind flux. For the momentum equation, the method is a new combined discontinuous Galerkin and finite element method approxima...

A detailed a priori error estimate is provided for a continuous-discontinuous Galerkin finite element method for the generalized 2D vorticity dynamics equations which describe several types of geophysical flows, including the Euler equations. The algorithm consists of a continuous Galerkin finite element method for the stream function and a discontinous Galerkin finite element method for the (p...