计算科学 - 近似∥ Δ f∥2大号2( Ω )‖Δf‖L2(Ω)2在有限元环境中 - 吾爱随笔录

我有形式的最小化问题

G (f) + ‖ Δ f ‖_{L^{2} (Ω)}^{2} \to min

$G(f) + \|\Delta f\|^2_{L^2(\Omega)} \to \min$ 在所有中，是封闭的和有界的。

f \in C^{2} (Ω)

$f\in C^2(\Omega)$

Ω

$\Omega$

让我们忘记 ; 我对如何离散化范数感兴趣。在有限体积的情况下，有所以其中是质量矩阵和投影到离散空间。 $G$ $L^2$

\begin{matrix} \int_{Ω} (Δ f)^{2} = \sum_{i} \int_{V_{i}} {(Δ f)}^{2} \approx \sum_{i} | V_{i} | (Δ f) (x_{i})^{2} \\ \approx \sum_{i} | V_{i} | {(| V_{i} |^{- 1} \int_{V_{i}} Δ f)}^{2} = \sum_{i} | V_{i} |^{- 1} {(\int_{V_{i}} Δ f)}^{2}, \end{matrix}

$\begin{multline} \int_\Omega (\Delta f)^2 = \sum_i \int_{V_i} \left(\Delta f\right)^2 \approx \sum_i |V_i| (\Delta f)(x_i)^2\\ \approx \sum_i |V_i| \left(|V_i|^{-1} \int_{V_i} \Delta f\right)^2 = \sum_i |V_i|^{-1} \left(\int_{V_i} \Delta f\right)^2, \end{multline}$

‖ Δ f ‖_{L^{2} (Ω)} \approx ‖ Δ_{h} P_{h} (f) ‖_{M^{- 1}}

$\|\Delta f\|_{L^2(\Omega)} \approx \|\Delta_h P_h(f)\|_{M^{-1}}$

M

$M$

P_{h}

$P_h$

从数值实验（见下文）看来，这也适用于有限元类型的函数（具有相应的质量矩阵；见下面的代码）。

有谁知道这是为什么？

import sympy
from dolfin import (
    Expression,
    FacetNormal,
    Function,
    FunctionSpace,
    TestFunction,
    TrialFunction,
    UnitSquareMesh,
    assemble,
    dot,
    ds,
    dx,
    grad,
    project,
    solve,
)

mesh = UnitSquareMesh(500, 500)
V = FunctionSpace(mesh, "CG", 1)

u = TrialFunction(V)
v = TestFunction(V)

n = FacetNormal(mesh)
A = assemble(dot(grad(u), grad(v)) * dx - dot(n, grad(u)) * v * ds)
M = assemble(u * v * dx)

f = Expression("sin(pi * x[0]) * sin(pi * x[1])", element=V.ufl_element())
x = project(f, V)

Ax = A * x.vector()
Minv_Ax = Function(V).vector()
solve(M, Minv_Ax, Ax)
val = Ax.inner(Minv_Ax)

print(val)


# Exact value
x = sympy.Symbol("x")
y = sympy.Symbol("y")
f = sympy.sin(sympy.pi * x) * sympy.sin(sympy.pi * y)
f2 = -sympy.diff(f, x, x) - sympy.diff(f, y, y)
val2 = sympy.integrate(sympy.integrate(f2 ** 2, (x, 0, 1)), (y, 0, 1))
print(sympy.N(val2))

输出：

97.75031146783857
97.4090910340024