In this paper, we considered the improved element-free Galerkin (IEFG) method for solving 2D anisotropic steady-state heat conduction problems. The moving least-squares (IMLS) approximation is used to establish trial function, and penalty applied enforce boundary conditions, thus final discretized equations of IEFG problems can be obtained by combining with corresponding weak form. influences n...

Discontinuous Galerkin (DG) and matrix-free finite element methods with a novel projective pressure estimation are combined to enable the numerical modeling of magma dynamics in 2D and 3D using the library deal.II . The physical model is an advection-reaction type system consisting of two hyperbolic equations to evolve porosity and soluble mineral abundance and one elliptic equation to recover ...

This paper presents a Galerkin-based meshless method for calculating stress-intensity factors (SIFs) for a stationary crack in two-dimensional functionally graded materials of arbitrary geometry. The method involves an element-free Galerkin method (EFGM), where the material properties are smooth functions of spatial coordinates and two newly developed interaction integrals for mixed-mode fractu...

saccharomyces cerevisiae (baker’s yeast) can exhibit sustained oscillations during the operation in a continuous bioreactor that adversely affects its stability and productivity. because of heterogeneous nature of cell populations, the cell population balance equation (pbe) can be used to capture the dynamic behavior of such cultures. in this work, an unstructured-segregated model is used for d...

We present an overview of hybridizable discontinuous Galerkin methods for solving partial differential equations (PDEs) in continuum mechanics [1, 2, 3, 4]. The essential ingredients are a local Galerkin projection of the underlying PDEs at the element level onto spaces of polynomials of degree k to parametrize the numerical solution in terms of the numerical trace; a judicious choice of the nu...

DarcyLite is a Matlab toolbox for numerical simulations of flow and transport in 2-dim porous media. This paper focuses on the finite element (FE) methods and the corresponding code modules in DarcyLite for solving the Darcy equation. Specifically, four major types of finite element solvers, the continuous Galerkin (CG), the discontinuous Galerkin (DG), the weak Galerkin (WG), and the mixed fin...

We introduce and analyze a discontinuous Galerkin method for the incompressible Navier-Stokes equations that is based on finite element spaces of the same polynomial order for the approximation of the velocity and the pressure. Stability of this equal-order approach is ensured by a pressure stabilization term. A simple element-by-element postprocessing procedure is used to provide globally dive...

(1) A one-dimensional minimization problem and the Ritz method. (2) Weak formulation and the Galerkin method. Abstract error estimates. (3) The lemma of variations. Euler-Lagrange equations. Weak formulation, again. (4) The Ritz-Galerkin finite element method: philosophy. (5) The piecewise linear finite element space and basis functions. The linear system. (6) The piecewise linear finite elemen...

Over the last few years, several mixed discontinuous Galerkin finite element methods (DGFEM) have been proposed for the discretization of incompressible fluid flow problems. We mention here only the piecewise solenoidal discontinuous Galerkin methods introduced in [5,25], the local discontinuous Galerkin methods of [12,11], and the interior penalty methods studied in [24,33,18]. Some of the mai...

Proliferation of degrees-of-freedom has plagued discontinuous Galerkin methodology from its inception over 30 years ago. This paper develops a new computational formulation that combines the advantages of discontinuous Galerkin methods with the data structure of their continuous Galerkin counterparts. The new method uses local, element-wise problems to project a continuous finite element space ...