Xem các phần khác tại đây.

Divergence theorem (Gauss’s theorem)

More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region. From wiki.

$$\iiint _{V}\left(\mathbf {\nabla } \cdot \mathbf {F} \right)\,dV= \int {\displaystyle \scriptstyle S} \scriptstyle S {\displaystyle (\mathbf {F} \cdot \mathbf {n} )\,dS.} (\mathbf {F} \cdot \mathbf {n} )\,dS.$$

---

Homeomorphic (đồng phôi) = Continuous, one-to-one, in surjection, and having a continuous inverse. (Liên tục + song ánh + ánh xạ ngược cũng liên tục)

---

Bolzano–Weierstrass theorem : Every each bounded sequence in $\mathbb{R}^n$ has a convergent subsequence.

Mọi dãy bị chặn đều có dãy con hội tụ.

Convergence theo nghĩa thông thường $\Vert u-u_h \Vert$ chỉ có thể áp dụng nếu như chứng minh được nghiệm duy nhất. Còn không thì phải làm theo hướng hội tụ yếu hay hầu khắp nơi gì đó.

Rellich

Rellich (relative compact, file analysis/rellich relative compact.pdf) Let $X,Y$ be two normed linear spaces, suppose $T:X \to Y$ is a linear operator. The the following are equivalent,

1. $T$ is compact. ($T$ is called a compact operator if for all bounded sets $E\subseteq X$, $T(E)$ is relative compact in $Y$ and $T$ is continuous.)
2. The image of the unit ball under $T$ is relative compact in $Y$.
3. For any bounded sequence ${x_n}$ in $X$, there exist a subsequence ${Tx_{x_k}}$ of ${Tx_n}$ that converges (strongly) in $Y$. $\Rightarrow$ Cái này cũng suy ra chỗ nếu dãy bị chặn trong $H^1 \hookrightarrow L^2$ thì tồn tại dãy con hội tụ mạnh trong $L^2$ luôn, dãy con đó chỉ hội tụ yếu trong $H^1$ mà thôi.

---

Rellich’s properties (file analysis/rellich relative compact.pdf)

• Any bounded set in $H^1_0(\Omega)$ is relative compact in $L^2(\Omega)$.
• Any bounded sequence in $H^1_0(\Omega)$ has a subsequence that converges strongly in $L^2(\Omega)$.
• Compact operator maps quickly convergence to strongly convergence. (file weak weakstar convergence.pdf)

---

Rellich-Kondrachov : Let $\Omega$ be a regular bounded domain in $\mathbb{R}^n$

• Suppose that $1\le p < n $ and $ 1 \le q < p^* $, then bounded set in $W^{1,p}(\Omega)$ is precompact (relatively compact) in $L^q(\Omega)$
• Suppose that $p>n$ then bounded sets in $W^{1,p}(\Omega)$ is precompact (relatively compact) in $C(\overline{\Omega})$.

---

Rellich-Kondrachov (Duong Minh Duc) : Let $D$ be a bounded open subset with smooth boundary in $\mathbb{R}^n$, $k$ be positive integer, and $p\in [1,\infty)$ such that $kp<n$. Let $q\in [1;\frac{np}{n-kp})$ and put

$$T(u)=u, \forall u\in W^{k,p}(D)$$

Then $T$ is a bounded linear mapping from $W^{k,p}(D)\to L^q(D)$ and $\overline{T(A)}$ is compact in $L^q(D)$ for any bounded subset $A\subset W^{1,p}(D)$.

Cũng trong file của thầy Đức, định lý trên cũng đúng cho $p\in (1,\infty), q\in [1,\infty)$.

Inequality

Poincaré inequality, equivalent norms chuẩn $\Vert{\nabla u}\Vert_ {L^p(\Omega)} \sim \Vert{u}\Vert_ {W^{1,p}(\Omega)}$ (cf. stackexchange)

Poincare inequality : $\Vert{u}\Vert_{L^2(\Omega)} \le C \Vert{\nabla u}\Vert_{L^2(\Omega)}$

---

Growall inequality : xem một phiên bản trong sách của Alexandre Ern (cf. Ern2004, chapter 6).

Theorems

Banach’s fixed point theorem : có thể xem Theorem 8.4 p.345 của Knabner2002 hoặc section 2.1 của Vasile2007

---

Let $U, W$ be subspaces of $V$ . Then $V = U \oplus W$ if and only if for every $v \in V$ there exist unique vectors $u \in U$ and $w \in W$ such that $v = u + w$.

---

Riesz Representation Theorem:

If $T$ is a bounded linear functional on a Hilbert space $H$ then there exists some $g \in H$ such that for every $f \in H$ we have $T(f)=\left<{f,g}\right>$ Moreover, $\left|{T}\right|=\left|{g}\right|$ (operator norm of $T$ and Hilbert space norm of $g$)

---

Second Strang Lemma: Let $u\in V, u_h\in V_h$ be arbitrary. Then $\begin{align*} \left\|{u-u_h}\right\|_{a} \le \text{inf}\left\|{u-v_h}\right\|_{v_h}+\text{sup}_{v_h}\dfrac{a(u-u_h,v_h)}{\left\|{v_h}\right\|_{a}} \end{align*}$ where $\left|{v}\right|_{a}=\sqrt{a(v,v)}$.

---

The “fundamental lemma of the calculus of variations” or “Du Bois-Reymond’s lemma” : Lemma 3.2 trong link này (chương 2 thì xem ở đây )và các biến thể của nó. Dưới đây là phát biểu sơ lược lại,

When $f\in L_{1,loc}(\Omega)$ with $\int f(x)\varphi(x)=0$ for all $\varphi\in C^{\infty}_0$ then $f=0$.

---

Theorem 2.7 (p.56 pdf in LeDret2013]) Soit $E$ un espace euclidien (donc de dimension finie) et soit $P:E \to E$ une application continue telle qu’il existe $\rho>0$ pour lequel tout point $x$ sur la sphère de rayon $\rho$ satisfait $P(x)\cdot x \ge 0$. Il existe alors un point $x_0, \left|{x_0}\right|_{}\le \rho$, tel que $P(x_0)=0$. $\Leftarrow$ Cái này dùng để áp dụng định lý fixed-point theorem chứng minh phương trình nonlinear có nghiệm nè.

Không phân loại

Support of a function $f: X\to \mathbb{R}$ is

$$\operatorname {supp} (f)=\{x\in X\,|\,f(x)\neq 0\}.$$

Smallest subset of $X$. If f(x) = 0 for all but a finite number of points x in X, then f is said to have finite support.

---

Scaling argument : Xem sách “A Posteriori Error Estimation in Finite Element Analysis” của Mark Ainsworth và J. Tinsley Oden. Trong đây cũng có kết quả khá hay (trang 8 của sách), dẫn đến cái sau đây $\Vert v_h\Vert_{L^p(\partial\Omega)} \le Ch^{-1/p}\Vert v_h\Vert_{L^p(\Omega)}$

---

Caratheodory function (cf. Kubinska2005) : Let $X, Y$ be topological spaces and $(T, M, \mu)$ be a measurable space. We say that $f : T \times X \to Y$ is a Caratheodory function if

1. $f(t,\cdot)$ is continuous for each $t$.
2. $f(\cdot,u)$ is measurable for each $u$.

---

Direct sum : xem ở stackexchange.

Direct sum is a term for subspaces, while sum is defined for vectors. We can take the sum of subspaces, but then their intersection need not to be ${0}$.

Examples:

Let $u = (0,1),v = (1,0), w= (1,0)$,

Then:

• $u + v = (1,1)$ (sum of vectors),
• $\operatorname{span}(v)+\operatorname{span}(w) = \operatorname{span}(v)$, so the sum is not direct,
• $\operatorname{span}(u)\oplus\operatorname{span}(v) = \Bbb R^2$, here the sum is direct because $\operatorname{span}(u)\cap\operatorname{span}(v)={0}$,
• $u \oplus v $ makes no sense in this context.

Còn đây là lý thuyết ở trang wikibook :

Where $W_1,W_2,\cdots,W_k$ are subspaces of a vector space, their sum is the span of their union $W_1+W_2+\cdots+W_k = [W_1 \cup W_2\cup\cdots\cup W_k ]$

---

Nonconformity : the discrete solution space $V^h$ is not a subspace of the continuous solution space $H^1_g(\Omega)$ - because the Dirichlet boundary conditions are enforced only in a weak sense (see more at nitscheIdea).

---

Let $E$ and $F$ two topological vector spaces, and $T\colon E\to F$ a linear map. If $E$ is finite dimensional, then $T$ is continuous.

Ánh xạ tuyến tính trong không gian hữu hạn chiều thì liên tục.

---

$1<p<\infty$ where $p’$ is the adjoint exponent with $1=\frac{1}{p} + \frac{1}{p’}$.

---

The jump and average:

• If $[v]=v_1-v_2$ and ${v} = \frac{1}{2}(v_1+v_2)$ then

  $$\begin{align*}[fg] &= \{f\}[g] + [f]\{g\}\\ \{fg\} &= \{f\}\{g\} + \frac{1}{2}[f][g]\end{align*}$$

• If $[v]=v_1-v_2$ and ${v} = \kappa_1 v_1+ \kappa_2 v_2, (\kappa_1+\kappa_2=1)$ then

  $$\begin{align*}[fg] &= \{f\}[g] + [f]\{g\} + (\kappa_1-\kappa_2)[f][g] \\ \{fg\} &= \{f\}\{g\} + \kappa_1\kappa_2 [f][g]\end{align*}$$

Jump and average : Xem thêm định nghĩa ở Definition 3.3.12 của interface/interface_lic_thesis.pdf, có định nghia theo lim.

---

Nonconformity : the discrete solution space $V^h$ is not a subspace of the continuous solution space $H^1_g(\Omega)$ - because the Dirichlet boundary conditions are enforced only in a weak sense (see more at nitscheIdea).

---

Homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

---

Every orthogonal projection is continuous.

---

Dense set : a subset $A$ of a topological space $X$ is dense in $X$ if for any point $x$ in $X$, any neighborhood of $x$ contains at least one point from $A$. $A$ dense in $X$ if $\bar{A}=X$.

---

Maximum principle : In mathematics, the maximum principle is a property of solutions to certain partial differential equations, of the elliptic and parabolic types. Roughly speaking, it says that the maximum of a function in a domain is to be found on the boundary of that domain.

Specifically, the strong maximum principle says that if a function achieves its maximumin the interior of the domain, the function is uniformly a constant.

The weak maximum principle says that the maximum of the function is to be found on the boundary, but may re-occur in the interior as well. Other, even weaker maximum principles exist which merely bound a function in terms of its maximum on the boundary.

Maximum principle cũng xem ở trong Ern2004 chapter 6. Theory and Practice of Finite Elements

Có thể xem thêm ở đây (Chapter 3).

---

Ý nghĩa tượng trưng của Lipschitz continuity (cf. background/lipschitz continuity.pdf) When we graph a function $f(x)$ of a rational variable $x$, we make a leap of faith and assume that the function values $f(x)$ vary “smoothly” or “continuously” between the sample points $x$, so that we can draw the graph of the function without lifting the pen. In particular, we assume that the function value $f(x)$ does not make unknown sudden jumps for some values of $x$. We thus assume that the function value $f(x)$ changes by a small amount if we change $x$ by a small amount. A basic problem in Calculus is to measure how much the function values $f(x)$ may change when $x$ changes, that is, to measure the “degree of continuity” of a function.

---

Monotone increasing : Always increasing; never remaining constant or decreasing. Alsocalled strictly increasing.

---

Stokes vs Navier-Stokes equation (cái này biết được trong lần đi dự hội nghị Toán Sinh tại trường Paris 13 từ một bài báo cáo của thằng kia)

• Stokes equation

$$\begin{align*}-\mu \Delta u + \nabla p &= \rho f \quad (\text{in }\Omega) \\ \text{div}(u) &= 0 \quad (\text{in }\Omega)\\ u &= 0 \quad (\text{on }\partial\Omega)\end{align*}$$

• Navier-Stokes equation

  $$\begin{align*}\frac{\partial}{\partial_t}(\rho u) +\text{div}(\rho u \otimes u) -\mu\Delta u + \nabla p &= \rho f \quad (\text{in }\Omega) \\ \text{div}(u) &= 0 \quad (\text{in }\Omega)\\ u &= 0 \quad (\text{on }\partial\Omega)\end{align*}$$

---

Conservation law : xem mục 5.1 trong sách Hesthaven2008. Nếu trong fluid dynamics thì xem trên wikipedia có vẻ rõ hơn.

---

Eigenvector and eigenvalue : In linear algebra, an eigenvector or characteristic vector of a square matrix is a vector that does not change its direction under the associated linear transformation. $Av=\lambda v$ ($\lambda$ is eigenvalue of square matrix $A$). Solving $(A-\lambda I)v=0$

---

Self-adjoint operator : (ý tưởng cơ bản, chưa hoàn toàn đầy đủ) linear map $A$ from $V$ to itself has $\left<{Av,w}\right> = \left<{v,Aw}\right>$.

---

Well-posed problem : The mathematical term well-posed problem stems from a definition given by Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that

• A solution exists.
• The solution is unique.
• The solution’s behavior changes continuously with the initial conditions.

---

$L^2(\Omega)$-orthogonal projection onto $V_h \subset L^2(\Omega)$. See more on page 19 of ErnDiscont

---

Discrete stability : có liên quan đến $a_h$ See definition 1.2.8

---

Cái cách chứng minh continuos $\vert f(x) \vert \le C \Vert x \Vert$ thì chỉ áp dụng được đối với $f$ linear/bilinear thôi hà. Lúc mình chứng minh hàm $G_h(u,v)$ mà có chứa 1 hạng tử nonlinear, áp dụng cái này bị cô la và bảo không áp dụng được.

---

$\nabla\cdot(uB) = (\nabla \cdot u)B + u\cdot \nabla B​$ : Nếu ghi cụ thể lý thuyết ra thì sẽ chứng minh được cái này đúng. Chắc chắn!

---

Một số cái biến đổi tích phân từng phần dùng trong chứng minh biểu thức có liên quan đến vận tốc.

$$\int_{\Omega}(u\cdot\nabla B)w = -\int_{\Omega}(\nabla\cdot u)Bw - \int_{\Omega}(Bu)\cdot\nabla w + \int_{\partial\Omega}(u\cdot n)Bw.$$

Cái trên xuất phát từ

$$\int_{\Omega}\nabla\cdot(uB)v = \int_{\Omega}\left(u\cdot\nabla B + (\nabla\cdot u)B\right)w = \int_{\partial\Omega}(u\cdot n)Bw - \int_{\Omega}(Bu)\cdot\nabla w.$$

---

$M=(A,d)$ is metric space, every convergent sequence in $M$ is a Cauchy sequence. (Mọi dãy hội tụ đều là dãy Cauchy). Xem chứng minh ở đây.

---

Ngược lại chưa chắc đúng. Nhưng nếu dãy số thực thì dãy Cauchy thì hội tụ (xem chứng minh ở đây, định ly 1), còn trong không gian tổng quát thì chưa chắc!

---

Lưu ý nếu trong không gian $M$, mọi dãy Cauchy mà hội tụ thì không gian $M$ được gọi là complete metric (không gian metric đầy đủ).

---

Equivalent norm : Hai chuẩn trong không gian hữu hạn chiều thì tương đương. Xem chứng minh trong note của

---

Posibivity: (p.285) và Maximum principle (p.286) cũng xem ở trong Ern2004, chapter 6.

Xem các phần khác tại đây.