有限元空间

1 Hilbert spaces

From a mathematical point of view, the "right" choice of function space is essential since this may make it easier to prove the existence of a solution to the continuous problem.
A weak formulation is obtained by multiplying the original equation by test functions and then integrating it. The advantage of the weak formulation is that it is easy to prove the existence of a solution.

Definition 1. Linear form.

If \(V\) is a linear space, we say that \(L\) is a linear form in \(V\) if \(L: V \to \mathbb{R}\), i.e., \(L(v) \in \mathbb{R}\) for \(v \in V\), and \(L\) is linear, i.e., for all \(v, w \in V\) and \(\beta, \theta \in \mathbb{R}\),

\[ L(\beta v + \theta w) = \beta L(v) + \theta L(w) \]

Definition 2. Bilinear form.

\(a(\cdot, \cdot)\) is a bilinear form on \(V \times V\) if \(a: V \times V \to \mathbb{R}\).

i.e., \(a(v, w) \in \mathbb{R}\) for \(v, w \in V\), and \(a(\cdot, \cdot)\) is linear in each argument, i.e., for all \(u, v, w \in V\) and \(\beta, \theta \in \mathbb{R}\), we have

\[ a(u, \beta v + \theta w) = \beta a(u, v) + \theta a(u, w) \]

\[ a(\beta u + \theta v, w) = \beta a(u, w) + \theta a(v, w) \]

The bilinear form \(a(\cdot, \cdot)\) is symmetric on \(V \times V\) if \(a(v, w) = a(w, v)\).

Definition 3. Scalar product.

A scalar product is a symmetric bilinear form \(a(\cdot, \cdot)\) on \(V \times V\) where \(a(\cdot, \cdot) > 0\) for all \(v \in V\), \(v \neq 0\).

Definition 4. Norm.

The norm \(\|\cdot\|_a\) associated with a scalar product \(a(\cdot, \cdot)\) is defined by

\[ \|v\|_a = [a(v, v)]^{1/2} \quad \forall v \in V \]

(bilinear product → scalar product → norm)

Here's your content in markdown format, with equations properly formatted:

Theorem 5. Cauchy's inequality.

If \(\langle \cdot, \cdot \rangle\) is a scalar product with corresponding norm \(\|\cdot\|\), then we have Cauchy's inequality: $$ |\langle v, w \rangle| \leq |v| |w| $$

Definition 6. Hilbert space.

If \(V\) is a linear space with a scalar product with corresponding norm \(\|\cdot\|\), then \(V\) is said to be a Hilbert space if \(V\) is complete, i.e., if every Cauchy sequence with respect to \(\|\cdot\|\) is convergent.

We could just think of a Hilbert space simply as a linear space with a scalar product (a special positive symmetric bilinear form).

Remark 7. Cauchy sequence.

A sequence \(v_1, v_2, v_3, \dots\) of elements \(v_i\) in the space \(V\) with norm \(\|\cdot\|\) is said to be a Cauchy sequence if for all \(\epsilon > 0\), there is a natural number \(N\) such that \(\|v_i - v_j\| < \epsilon\) if \(i, j > N\). Further, \(v_i\) converges to \(v\) if \(\|v - v_i\| \to 0\) as \(i \to \infty\).

Definition 8. \(C^\infty_0(\Omega)\)

\(C^\infty_0(\Omega)\) is the set of all functions that are infinitely differentiable on \(\Omega\) and compactly supported in \(\Omega\).

Definition 9. \(L^p\) space (finite energy space)

\[ L^p(\Omega) = \left\{ v: \Omega \to \mathbb{R}: \int_\Omega |v|^p \, dx \, dy < \infty \right\} \]

Corollary 10. \(L^2\) space

\[ L^2(\Omega) = \left\{ v: \Omega \to \mathbb{R}: \int_{\Omega} v^2 \, dx \, dy < \infty \right\} \]

The \(L^2(\Omega)\) space is a Hilbert space with the scalar product

\[ (v, w) = \int_{\Omega} v w \, dx \]

and the corresponding norm (the \(L^2\)-norm):

\[ \|v\|_{L^2(\Omega)} = \left( \int_{\Omega} v^2 \, dx \right)^{1/2} = (v, v)^{1/2} \]

Definition 11. \(L^\infty\) space

\[ L^\infty(\Omega) = \left\{ v: \Omega \to \mathbb{R}: \sup_{\Omega} |v| < \infty \right\} \]

Definition 12. \(H^m(\Omega)\) space

\[ H^m(\Omega) = \left\{ v \in L^2(\Omega): \frac{\partial^{\alpha_1 + \alpha_2} v}{\partial x^{\alpha_1} \partial y^{\alpha_2}} \in L^2(\Omega), \forall \alpha_1 + \alpha_2 = 1, \dots, m \right\} \]

Corollary 13. \(H^1(\Omega)\) space (where \(v\) and \(v'\) belong to \(L^2(\Omega)\))

\[ H^1(\Omega) = \left\{ v \in L^2(\Omega): \frac{\partial^{\alpha_1 + \alpha_2} v}{\partial x^{\alpha_1} \partial y^{\alpha_2}} \in L^2(\Omega), \forall \alpha_1 + \alpha_2 = 1 \right\} \]

The \(H^1(\Omega)\) space has the scalar product:

\[ (v, w)_{H^1(\Omega)} = \int_{\Omega} (v w + v' w') \, dx \]

and the corresponding norm:

\[ \|v\|_{H^1(\Omega)} = \left( \int_{\Omega} \left(v^2 + (v')^2\right) \, d\Omega \right)^{1/2} \]

Definition 14. \(H^1_0(\Omega)\) space

\[ H^1_0(\Omega) = \left\{ v \in H^1(\Omega): v = 0 \text{ on } \partial \Omega \right\} \]

Definition 15. \(W^m_p(\Omega)\) space

\[ W^m_p(\Omega) = \left\{ v: \Omega \to \mathbb{R}: \int_{\Omega} \left[\frac{\partial^{\alpha_1 + \alpha_2} v}{\partial x^{\alpha_1} \partial y^{\alpha_2}}\right]^p \, dx \, dy < \infty, \forall \alpha_1 + \alpha_2 = 0, \dots, m \right\} \]

Remark 16.

\[ L^p(\Omega) = W^0_p(\Omega); \]

\[ L^2(\Omega) = W^0_2(\Omega); \]

\[ H^m(\Omega) = W^m_2(\Omega); \]

\[ H^1(\Omega) = W^1_2(\Omega) \]

有限元误差分析

通常 \(L^\infty\) 范数误差、\(L^2\) 范数误差和 \(H^1\) 半范数误差是三种常用的误差度量标准。

• \(L^\infty\) (L-infinity Norm Error): 峰值误差， 关注局部最坏情况， 衡量的是在整个求解域误差函数的绝对值的最大值。

\[ \|e\|_{L^\infty(\Omega)} = \|u - u_h\|_{L^\infty(\Omega)} = \sup_{x \in \Omega} |u(x) - u_h(x)| \]

其中 sup 指的是上确界，对于连续函数而言是最大值。

• \(L^2\) (L-2 Norm Error): 均方根误差， 测量的是误差函数在整个求解域 \(\Omega\) 内的“平均”大小

\[ \|e\|_{L^2(\Omega)} = \|u - u_h\|_{L^2(\Omega)} = \left( \int_{\Omega} (u(x) - u_h(x))^2 \, d\Omega \right)^{1/2} \]

• \(H^1\) 半范数误差 (H-1 Semi-norm Error): 导数误差， 衡量的是近似解的梯度与真实解的梯度之间的 \(L^2\) 误差

\[ |e|_{H^1(\Omega)} = |u - u_h|_{H^1(\Omega)} = \left( \int_{\Omega} |\nabla(u(x) - u_h(x))|^2 \, d\Omega \right)^{1/2} \]

在有限元收敛性分析中，我们通常会看到这三种误差随着网格加密（\(h \to 0\)）而减小。对于使用 \(p\) 次多项式的有限元单元，理论上可以得到如下的收敛速度估计：

• \(H^1\) 半范数误差: \(\|u-u_h\|_{H^1} = O(h^p)\)
• \(L^2\) 范数误差: \(\|u-u_h\|_{L^2} = O(h^{p+1})\)
• \(L^\infty\) 范数误差: \(\|u-u_h\|_{L^\infty}\) 的收敛阶通常与 \(L^2\) 误差相似，但可能稍差一些，有时会包含一个对数项，如 \(O(h^{p+1}|\ln h|)\)。

\(L^2\) 范数（函数值本身）的收敛速度通常比 \(H^1\) 半范数（导数值）的收敛速度快一个阶次。这符合直觉：要准确地逼近一个函数的变化趋势（导数）比仅仅逼近它的函数值要更困难一些。