scholarly journals Arbitrary-Order Bernstein–Bézier Functions for DGFEM Transport on 3D Polygonal Grids

2021 ◽  
Vol 2 (3) ◽  
pp. 239-245
Author(s):  
Michael Hackemack
Keyword(s):  
Convergence Rates ◽  
Arbitrary Order ◽  
Diffusion Limit ◽  
Element Discretization ◽  
Lagrange Polynomials ◽  
Galerkin Finite Element ◽  
Polynomial Degree ◽  
Numerical Tests ◽  
Full Resolution ◽  
Discontinuous Galerkin Finite Element

In this paper, we present an arbitrary-order discontinuous Galerkin finite element discretization of the SN transport equation on 3D extruded polygonal prisms. Basis functions are formed by the tensor product of 2D polygonal Bernstein–Bézier functions and 1D Lagrange polynomials. For a polynomial degree p, these functions span {xayb}(a+b)≤p⊗{zc}c∈(0,p) with a dimension of np(p+1)+(p+1)(p−1)(p−2)/2 on an extruded n-gon. Numerical tests confirm that the functions capture exactly monomial solutions, achieve expected convergence rates, and provide full resolution in the thick diffusion limit.

scholarly journals Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

2015 ◽  
Vol 17 (3) ◽  
pp. 721-760 ◽  
Author(s):  
Arnaud Duran ◽  
Fabien Marche
Keyword(s):  
Discontinuous Galerkin ◽  
Numerical Procedure ◽  
Computationally Efficient ◽  
Approximation Space ◽  
Galerkin Finite Element ◽  
Polynomial Degree ◽  
New Class ◽  
New Family ◽  
Discontinuous Galerkin Finite Element ◽  
Water Height

AbstractWe describe in this work a discontinuous-Galerkin Finite-Element method to approximate the solutions of a new family of 1d Green-Naghdi models. These new models are shown to be more computationally efficient, while being asymptotically equivalent to the initial formulation with regard to the shallowness parameter. Using the free surface instead of the water height as a conservative variable, the models are recasted under apre-balancedformulation and discretized using a nodal expansion basis. Independently from the polynomial degree in the approximation space, the preservation of the motionless steady-states is automatically ensured, and the water height positivity is enforced. A simple numerical procedure devoted to stabilize the computations in the vicinity of broken waves is also described. The validity of the resulting model is assessed through extensive numerical validations.

scholarly journals Energy Stable Interior Penalty Discontinuous Galerkin Finite Element Method for Cahn–Hilliard Equation

2017 ◽  
Vol 18 (5) ◽  
pp. 303-314 ◽  
Author(s):  
Ayşe Sarıaydın-Filibelioğlu ◽  
Bülent Karasözen ◽  
Murat Uzunca
Keyword(s):  
Discontinuous Galerkin ◽  
Convergence Rates ◽  
Logarithmic Potential ◽  
Galerkin Finite Element Method ◽  
Interior Penalty ◽  
Galerkin Finite Element ◽  
Degenerate Mobility ◽  
Discontinuous Galerkin Finite Element ◽  
Energy Stable ◽  
Cahn Hilliard Equation

AbstractAn energy stable conservative method is developed for the Cahn–Hilliard (CH) equation with the degenerate mobility. The CH equation is discretized in space with the mass conserving symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting semi-discrete nonlinear system of ordinary differential equations are solved in time by the unconditionally energy stable average vector field (AVF) method. We prove that the AVF method preserves the energy decreasing property of the fully discretized CH equation. Numerical results for the quartic double-well and the logarithmic potential functions with constant and degenerate mobility confirm the theoretical convergence rates, accuracy and the performance of the proposed approach.

scholarly journals Performance of Adaptive Unstructured Mesh Modelling in Idealized Advection Cases over Steep Terrains

Atmosphere ◽  
2018 ◽  
Vol 9 (11) ◽  
pp. 444 ◽  
Author(s):  
Jinxi Li ◽  
Jie Zheng ◽  
Jiang Zhu ◽  
Fangxin Fang ◽  
Christopher. Pain ◽  
...  
Keyword(s):  
Unstructured Mesh ◽  
Computational Cost ◽  
Adaptive Mesh ◽  
Galerkin Finite Element Method ◽  
Adaptive Meshes ◽  
Galerkin Finite Element ◽  
Computational Costs ◽  
Cut Cell ◽  
Discontinuous Galerkin Finite Element ◽  
Terrain Following

Advection errors are common in basic terrain-following (TF) coordinates. Numerous methods, including the hybrid TF coordinate and smoothing vertical layers, have been proposed to reduce the advection errors. Advection errors are affected by the directions of velocity fields and the complexity of the terrain. In this study, an unstructured adaptive mesh together with the discontinuous Galerkin finite element method is employed to reduce advection errors over steep terrains. To test the capability of adaptive meshes, five two-dimensional (2D) idealized tests are conducted. Then, the results of adaptive meshes are compared with those of cut-cell and TF meshes. The results show that using adaptive meshes reduces the advection errors by one to two orders of magnitude compared to the cut-cell and TF meshes regardless of variations in velocity directions or terrain complexity. Furthermore, adaptive meshes can reduce the advection errors when the tracer moves tangentially along the terrain surface and allows the terrain to be represented without incurring in severe dispersion. Finally, the computational cost is analyzed. To achieve a given tagging criterion level, the adaptive mesh requires fewer nodes, smaller minimum mesh sizes, less runtime and lower proportion between the node numbers used for resolving the tracer and each wavelength than cut-cell and TF meshes, thus reducing the computational costs.

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