离散伽辽金方法：一种数值计算技术
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"DG_Method PDF文件，由Bernardo Cockburn撰写，探讨了离散伽辽金方法在科学计算中的应用，主要来源于教学和研究机构，发布于HAL开放存取档案，遵循Creative Commons Attribution 4.0 International License。该文档发表于2003年的《应用数学与力学》杂志，详细阐述了离散伽辽金方法的理论和实践，包括在解决偏微分方程中的应用。"
离散伽辽金方法（Discontinuous Galerkin Method，简称DG方法）是一种用于数值求解偏微分方程（PDEs）的高级有限元素方法。这种方法最初由Ronald Rivlin和Reinhard Stenberg在1970年代提出，后来由Bernardo Cockburn等人进一步发展和完善。DG方法的特点在于它允许在元素内部和元素之间的函数是不连续的，这与传统的连续有限元素方法形成了鲜明对比。
在DG方法中，域被划分为多个互不重叠的子区域（或称为元素），每个子区域内的解可以是任意的，但通常选择为多项式形式。在元素边界上，DG方法通过特定的接口条件来连接这些不连续的解，这些条件可以是弱形式的边界条件或者通过数值通量来实现。这种方法的一个显著优势是它具有高度的灵活性，可以适应复杂的几何形状和网格划分，同时也能处理非匹配网格。
DG方法的另一个优点是它的局部性和高阶精度。由于解在元素内部可以自由变化，因此可以很容易地实现高阶精度，这对于需要高精度模拟的问题非常有用。此外，由于每个元素的解是独立的，DG方法也适合并行计算，这在处理大规模问题时具有显著优势。
在实际应用中，DG方法广泛应用于流体力学、固体力学、电磁学、热传导等领域的偏微分方程求解。例如，在流体力学中，DG方法可以用来求解纳维-斯托克斯方程；在固体力学中，它可以解决弹性动力学问题；在电磁学中，DG方法则用于求解麦克斯韦方程。
Bernardo Cockburn的这篇论文详细介绍了离散伽辽金方法的基本概念、理论框架以及具体实现步骤，并提供了相关的数值实验来验证方法的性能和稳定性。论文中还可能涵盖了误差分析、选择适当的数值通量、以及如何调整方法参数以优化求解效率等内容。
离散伽辽金方法是一种强大而灵活的数值方法，它在科学和工程计算中有着广泛的应用，并且随着计算硬件的发展和并行计算技术的进步，其重要性将继续增加。这篇由Bernardo Cockburn撰写的论文是理解并掌握DG方法的一个重要参考资源。
jumps. We also comment on some computational issues and establish various links with other numerical schemes. In Sect. 2,
devoted to linear problems, we consider (i) the transport equation, (ii) the wave equation, (iii) the Poisson equation, and (iv)
the Oseen system. In Sect. 3, we consider the more difficult non-linear hyperbolic problems. We end the paper in Sect. 4, with
some concluding remarks.
2 Linear problems
2.1 The transport equation
In this sub-section, we consider DG methods for the transport equation
u
t
+ ∇·(a u)=0 in R
d
× (0,T),
u(t =0)=u
0
on R
d
.
We consider only the discretization of this equation in space; the full discretization will be studied when when deal with
non-linear hyperbolic problems in Sect. 3.
The objective here is to examine three properties that are especially relevant in this case. The first is that when the DG
methods are strongly related to classical finite volume methods like the up-winding and the Lax-Friedrichs methods. The
second is that when polynomials of high degree are used, the DG method can achieve high-order accuracy while remaining
highly parallelizable. The third is that the artificial viscosity of the method is given by the size of the jumps which, in turn,
are associated with the residual inside the elements. As a consequence, as the polynomial degree of its approximate solution
increases, the artificial viscosity diminishes even in the presence of discontinuities.
The DG methods. To discretize the transport equation in space by using a DG method, we first triangulate the domain R
d
;
let us denote by T
h
such triangulation. We then seek a discontinuous approximate solution u
h
which, in each element K of
the triangulation T
h
, belongs to the space V (K). There is no restriction in how to choose the space V (K), although a typical
choice is the space of polynomials of degree at most k, P
k
(K). We determine the approximate solution on the element K by
weakly enforcing the transport equation as follows:
K
(u
h
)
t
v −
K
a u
h
·∇v +
∂K
a u
h
· n vds=0, (4)
for all v ∈ V (K). To complete the definition of the DG method, it only remains to define the numerical trace
a u
h
.
To do that, we proceed as in the ODE considered in the Introduction and begin by obtaining a stability result for the problem
under consideration. We thus multiply the transport equation by u, and integrate over the space and time to get
1
2
R
d
u
2
(x,T) dx +
1
2
T
0
R
d
∇·a(x) u
2
(x,t) dx dt =
1
2
R
d
u
2
0
(x) dx.
From this equation, a stability result immediately follows if we assume, for example, that −∇ · a ≤ L.
Now, let us mimic the above procedure for the DG method under consideration. Taking v = u
h
in the weak formulation
defining the approximate solution u
h
, and adding on the elements K,weget
1
2
R
d
u
2
h
(x,T) dx +
1
2
T
0
R
d
∇·a(x) u
2
h
(x,t) dx dt +
T
0
Θ
h
(t) dt =
1
2
R
d
u
2
h,0
(x) dx,
where
Θ
h
(t)=
K∈T
h
−
1
2
K
∇·(a u
h
)(x,t) dx +
∂K
a u
h
(x,t) · n u
h
(x,t) ds
.
Next, we investigate if it is possible to define the numerical trace
a u
h
in such a way as to render Θ
h
non-negative.
At this point, it is convenient to introduce the following notation. Let x be a point on the set e =
∂K
+
∩ ∂K
−
and let n
±
denote the unit outward normal to ∂K
±
at the point x. Let u
±
h
(x) denote the value lim
↓0
u
h
x − n
±
and set
{u
h
} =
1
2
u
+
h
+ u
−
h
, [[ u
h
]] = u
−
h
n
−
+ u
+
h
n
+
.
Finally, let E
h
denote the set of sets e = ∂K
+
∩ ∂K
−
for all K
+
and K
−
∈T
h
.
4
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