Multiple-Stencil Dispersion Analysis of the Lagrange Multipliers in a Discontinuous Galerkin Method for the Helmholtz Equation

2003 ◽  
Vol 11 (02) ◽  
pp. 239-254 ◽  
Author(s):  
Isaac Harari ◽  
Charbel Farhat ◽  
Ulrich Hetmaniuk
Keyword(s):  
Helmholtz Equation ◽  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Lagrange Multiplier ◽  
Lagrange Multipliers ◽  
Degrees Of Freedom ◽  
Dispersion Analysis ◽  
Dispersion Properties

We analyze the dispersion properties of elements obtained by a discontinuous Galerkin method with Lagrange multipliers. The dispersion analysis of these elements presents a challenge in that the Lagrange multiplier degrees of freedom are directional, and hence an unbounded mesh is made up of more than one repeating pattern. Two approaches to overcome this difficulty are presented. The similarity in the two sets of results offers mutual validation of the two approaches.

scholarly journals Dispersion analysis of the discontinuous Galerkin method as applied to the equations of dynamic elasticity theory

2015 ◽  
pp. 387-406
Author(s):  
В.В. Лисица
Keyword(s):  
Galerkin Method ◽  
Elasticity Theory ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Computational Cost ◽  
Dispersion Analysis ◽  
Numerical Dispersion ◽  
Seismic Modeling ◽  
Dynamic Elasticity ◽  
Very High

Приводится дисперсионный анализ разрывного метода Галеркина в применении к системе уравнений динамической теории упругости. В зависимости от степени базисных полиномов рассматриваются P1-, P2- и P3-формулировки метода при использовании регулярной треугольной сетки. Показано, что для задач сейсмического моделирования оптимальной является P2-формулировка, поскольку сочетает в себе достаточную точность (численная дисперсия не выше 0.05% и вычислительную эффективность. Использование P1-формулировки приводит к недопустимо высокой численной дисперсии, в то время как P3-формулировка является чрезвычайно ресурсоемкой при использовании дискретизаций от 3 до 20 ячеек сетки на длину волны, типичной для сейсмического моделирования. The dispersion analysis of the discontinuous Galerkin method as applied to the equations of dynamic elasticity theory is performed. Depending on the degrees of basis polynomials, we consider the P1, P2, and P3 formulations of this method in the case of regular triangular meshes. It is shown that, for the problems of seismic modeling, the P2 formulation is optimal, since a sufficient accuracy (the numerical dispersion does not exceed 0.05%) and the computational efficiency are achieved. The application of the P1 formulation leads to an undesirably high numerical dispersion. The P3 formulation allows one to obtain accurate results, but its computational cost is very high when the number of grid cells per wavelength belongs to range between 3 and 20, which is typical for the seismic modeling.

scholarly journals A Mixed Discontinuous Galerkin Method for the Helmholtz Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Qingjie Hu ◽  
Yinnian He ◽  
Tingting Li ◽  
Jing Wen
Keyword(s):  
Helmholtz Equation ◽  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Scattered Field ◽  
Numerical Experiments ◽  
Optimal Order ◽  
Flux Variable ◽  
Optimal Order Convergence ◽  
Element Pair

In this paper, we introduce and analyze a mixed discontinuous Galerkin method for the Helmholtz equation. The mixed discontinuous Galerkin method is designed by using a discontinuous Pp+1−1−Pp−1 finite element pair for the flux variable and the scattered field with p≥0. We can get optimal order convergence for the flux variable in both Hdiv-like norm and L2 norm and the scattered field in L2 norm numerically. Moreover, we conduct the numerical experiments on the Helmholtz equation with perturbation and the rectangular waveguide, which also demonstrate the good performance of the mixed discontinuous Galerkin method.

scholarly journals A contraction property of an adaptive divergence-conforming discontinuous Galerkin method for the Stokes problem

2018 ◽  
Vol 26 (4) ◽  
pp. 209-232 ◽  
Author(s):  
Natasha Sharma ◽  
Guido Kanschat
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Adaptive Algorithm ◽  
Degrees Of Freedom ◽  
Stokes Problem ◽  
Adaptive Divergence ◽  
Optimal Complexity ◽  
Contraction Property ◽  
Velocity Approximation

Abstract We prove the contraction property for two successive loops of the adaptive algorithm for the Stokes problem reducing the error of the velocity. The problem is discretized by a divergence-conforming discontinuous Galerkin method which separates pressure and velocity approximation due to its cochain property. This allows us to establish the quasi-orthogonality property which is crucial for the proof of the contraction. We also establish the quasi-optimal complexity of the adaptive algorithm in terms of the degrees of freedom.

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