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10,964
result(s) for
"Galerkin methods."
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Modeling shallow water flows using the discontinuous Galerkin method
Replacing the Traditional Physical Model Approach. Computational models offer promise in improving the modeling of shallow water flows. As new techniques are considered, the process continues to change and evolve. Modeling Shallow Water Flows Using the Discontinuous Galerkin Method examines a technique that focuses on hyperbolic conservation laws and includes one-dimensional and two-dimensional shallow water flows and pollutant transports. Combines the Advantages of Finite Volume and Finite Element Methods.
BOOK
Advances in the Improved Element-Free Galerkin Methods: A Comprehensive Review
The element-free Galerkin (EFG) method, which constructs shape functions via moving least squares (MLS) approximation, represents a fundamental and widely studied meshless method in numerical computation. Although it achieves high computational accuracy, the shape functions are more complex than those in the conventional finite element method (FEM), resulting in great computational requirements. Therefore, improving the computational efficiency of the EFG method represents an important research direction. This paper systematically reviews significant contributions from domestic and international scholars in advancing the EFG method. Including the improved element-free Galerkin (IEFG) method, various interpolating EFG methods, four distinct complex variable EFG methods, and a series of dimension splitting meshless methods. In the numerical examples, the effectiveness and efficiency of the three methods are validated by analyzing the solutions of the IEFG method for 3D steady-state anisotropic heat conduction, 3D elastoplasticity, and large deformation problems, as well as the performance of two-dimensional splitting meshless methods in solving the 3D Helmholtz equation.
Journal Article
Hybridizable discontinuous Galerkin methods for second-order elliptic problems: overview, a new result and open problems
We describe, in the framework of steady-state diffusion problems, the history of the development of the so-called hybridizable discontinuous Galerkin (HDG) methods, since their inception in 2009 until now. We show how it runs parallel to the development of the so-called hybridized mixed (HM) methods and how, a few years ago, it prompted the introduction of the
M
-decompositions as a novel tool for the construction of superconvergent HM and HDG methods for elements of quite general shapes. We then uncover a new link between HM and HDG methods, namely, that any HM method can be rewritten as an HDG method by a suitable transformation of a subspace of the approximate fluxes of the HM method into a stabilization function. We end by listing several open problems which are a direct consequence of this result.
Journal Article
hp$ -Version Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains
In this paper we introduce the $hp$-version discontinuous Galerkin composite finite element method for the discretization of second-order elliptic partial differential equations. This class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or microstructures. While standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features. The key idea in the construction of this latter class of methods is that the computational domain $\\Omega$ is no longer resolved by the mesh; instead, the finite element basis (or shape) functions are adapted to the geometric details present in $\\Omega$. In this paper, we extend these ideas to the discontinuous Galerkin setting, based on employing the $hp$-version of the finite element method. Numerical experiments highlighting the practical application of the proposed numerical scheme will be presented. [PUBLICATION ABSTRACT]
Journal Article
Analysis of a meshless method for the time fractional diffusion-wave equation
In this paper a numerical technique is proposed for solving the time fractional diffusion-wave equation. We obtain a time discrete scheme based on finite difference formula. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method and the convergence order of the time discrete scheme is
O
(
τ
3
−
α
)
. Firstly, we change the main problem based on Dirichlet boundary condition to a new problem based on Robin boundary condition and then, we consider a semi-discrete scheme with Robin boundary condition and show when
β
→
+
∞
solution of the main semi-discrete problem with Dirichlet boundary condition is convergent to the solution of the new semi-discrete problem with Robin boundary condition. We consider the new semi-discrete problem with Robin boundary condition and use the meshless Galerkin method to approximate the spatial derivatives. Finally, we obtain an error bound for the new problem. We prove that convergence order of the numerical scheme based on Galekin meshless is
O
(
h
)
. In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. The main aim of the current paper is to obtain an error estimate for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme.
Journal Article
A New Error Analysis of Crank–Nicolson Galerkin FEMs for a Generalized Nonlinear Schrödinger Equation
In this paper, we study linearized Crank–Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation. We present the optimal
L
2
error estimate without any time-step restrictions, while previous works always require certain conditions on time stepsize. A key to our analysis is an error splitting, in terms of the corresponding time-discrete system, with which the error is split into two parts, the temporal error and the spatial error. Since the spatial error is
τ
-independent, the numerical solution can be bounded in
L
∞
-norm by an inverse inequality unconditionally. Then, the optimal
L
2
error estimate can be obtained by a routine method. To confirm our theoretical analysis, numerical results in both two and three dimensional spaces are presented.
Journal Article
A computational study of the weak Galerkin method for second-order elliptic equations
The weak Galerkin finite element method is a novel numerical method that was first proposed and analyzed by Wang and Ye (
2011
) for general second order elliptic problems on triangular meshes. The goal of this paper is to conduct a computational investigation for the weak Galerkin method for various model problems with more general finite element partitions. The numerical results confirm the theory established in Wang and Ye (
2011
). The results also indicate that the weak Galerkin method is efficient, robust, and reliable in scientific computing.
Journal Article
A New Class of High-Order Energy Stable Flux Reconstruction Schemes
The flux reconstruction approach to high-order methods is robust, efficient, simple to implement, and allows various high-order schemes, such as the nodal discontinuous Galerkin method and the spectral difference method, to be cast within a single unifying framework. Utilizing a flux reconstruction formulation, it has been proved (for one-dimensional linear advection) that the spectral difference method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior flux collocation points are located at zeros of the corresponding Legendre polynomials. In this article the aforementioned result is extended in order to develop a new class of one-dimensional energy stable flux reconstruction schemes. The energy stable schemes are parameterized by a single scalar quantity, which if chosen judiciously leads to the recovery of various well known high-order methods (including a particular nodal discontinuous Galerkin method and a particular spectral difference method). The analysis offers significant insight into why certain flux reconstruction schemes are stable, whereas others are not. Also, from a practical standpoint, the analysis provides a simple prescription for implementing an infinite range of energy stable high-order methods via the intuitive flux reconstruction approach.
Journal Article
General DG-Methods for Highly Indefinite Helmholtz Problems
We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in
R
d
,
d
∈
{
1
,
2
,
3
}
. The theory covers conforming as well as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much weaker conditions. As an application we present the error analysis for the
hp
-version of the finite element method explicitly in terms of the mesh width
h
, polynomial degree
p
and wavenumber
k
. It is shown that the optimal convergence order estimate is obtained under the conditions that
kh
/
p
is sufficiently small and the polynomial degree
p
is at least
O
(
log
k
)
. On regular meshes, the first condition is improved to the requirement that
kh
/
p
be sufficiently small.
Journal Article
POLYNOMIAL-DEGREE-ROBUST A POSTERIORI ESTIMATES IN A UNIFIED SETTING FOR CONFORMING, NONCONFORMING, DISCONTINUOUS GALERKIN, AND MIXED DISCRETIZATIONS
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 respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme.
Journal Article