Stability and convergence of a finite element method for a semilinear elliptical problem with small viscosity

2019 ◽  
Vol 78 (10) ◽  
pp. 3363-3374
Author(s):  
Yarong Zhang ◽  
Yinnian He
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stability And Convergence ◽  
Small Viscosity ◽  
Element Method

scholarly journals Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation

2018 ◽  
Vol 16 (1) ◽  
pp. 1091-1103 ◽  
Author(s):  
Leilei Wei ◽  
Yundong Mu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Discontinuous Galerkin ◽  
Local Discontinuous Galerkin Method ◽  
Stability And Convergence ◽  
Lax Equation ◽  
Local Discontinuous Galerkin ◽  
Discontinuous Galerkin Finite Element ◽  
Numerical Performance ◽  
Element Method

AbstractIn this paper we develop and analyze the local discontinuous Galerkin (LDG) finite element method for solving the general Lax equation. The local discontinuous Galerkin method has the flexibility for arbitrary h and p adaptivity, and allows for hanging nodes. By choosing the numerical fluxes carefully we prove stability and give an error estimate. Finally some numerical examples are computed to show the convergence order and excellent numerical performance of proposed method.

Theoretical Investigation of Unconditionally Stable Sequential Algorithms of Coupled Flow and Geomechanics for Hydraulically Fractured Systems

2021 ◽  
Author(s):  
Jihoon Kim
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
A Priori ◽  
Coupled Systems ◽  
Stability And Convergence ◽  
Sequential Algorithms ◽  
Coupled Flow ◽  
Unconditionally Stable ◽  
Fracture Stiffness ◽  
Element Method

Abstract We investigate unconditionally stable sequential algorithms for coupled hydraulically fractured geomechanics and flow systems, which can account for poromechanics behavior within the fractures. We focus on modifying the concepts of the fixed stress and undrained sequential methods properly for the coupled systems by taking appropriate stabilization terms for stability and convergence with energy analyses. Specifically, an apparent fracture stiffness is used for for numerical stabilization. Because this fracture stiffness depends on the fracture length, the stabilization term needs to be updated dynamically, different from the drained bulk modulus used for typical poromechanics problems. For numerical tests, we take the extended finite element method for geomechanics while the piecewise constant finite element method is used for flow within an existing hydraulic fracture. The numerical results support a priori stability analyses.

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