Partial Differential Equation numerical solution --- learning Summary 1. Knowledge Review (Note: \ (\ mit V \) is a linear space)
- Inner Product $ (\ cdot, \ cdot): \ mit V \ times \ mit V \ longrightarrow \ r $ is a bilinear ing that satisfies \ (I) (u, v) = (V, u), \ forall \, U, V \ In \ mit V \);
? $ (II) (u, u) \ GE 0, \ forall, U \ In \ mit v $; \ (III) (u, u) = 0 \) only when \ (u = 0 \).
Semi-norm \ (| \ cdot |: \ mit V \ longrightarrow \ r \) is a linear ing that satisfies \ (I) | v | \ GE 0, \ forall \, V \ In \ mit V; \) $ (ii) | CV | |=| C ||| v ||, \ forall V \ In \ mit V, \ forall C \ In \ r; $
? \ (Iii) | u + v | \ Leq | u | + | v |, \ forall U, V \ In \ mit v .\)
Norm half-norm + condition: \ (| u | = 0 \) When and only when \ (u = 0 \).
Norm Equivalence Theorem: Set \ (| \ cdot | \) and \ (| \ cdot | \) to linear space \ (\ mathbf V \) if there are two positive numbers \ (C_1 \) and \ (C_2 ,\)
? If the following unequal expressions are met, \ (| \ cdot | \) and \ (| \ cdot | \) are equivalent,
\ [C_1 | v | \ Leq C_2 | v |, \ forall \, C_1, C_2 \ In \ r ,\, \ forall \, V \ In \ mit v. \]
Inner Product space if a linear space is given inner product, it is an inner product space.
Inner Product can generate induction norms
\ [| V | = (V, V) ^ {1/2}, \, \ forall \, V \ In \ mit v. \]
Schwarz Inequality
\ [| (W, V) | \ Leq | w | \, | v |, \, \ forall W, V \ In \ mit v. \]
- The complete inner product space of the Hilbert space is the Hilbert space, that is, any kernel sequence in the inner product space is converged.
- Norm linear space if a linear space is assigned a norm, it becomes norm linear space.
- The inner product space must be a norm linear space, and its norm is \ (| v | = (V, V) ^ {1/2}, \, \ forall \, V \ In \ mit v. \)
- The complete norm linear space in the he space, that is, any sequence of the keys in the norm linear space is converged.
- The Hilbert space must be the space of the East Coast.
2. New Concept
Dual Space
If (\ mit V, | \ cdot | _ {\ mit v} \) and (\ mit W, | \ cdot | _ {\ mit w} \) is two norm linear spaces, from \ (\ mathbf V \) to \ (\ mit w \) the linear function of constitutes a norm linear space, which is recorded as \ (\ scr l (\ mit V; \ mit W )\). for \ (L \ In \ scr l (\ mit V; \ mit W) \), the defined norms are as follows:
\ [| L | _ {\ scr l (\ mit V; \ mit W )}: = \ sup _ {0 \ neq v \ In \ mit v} \ frac {| LV |||_{\ mit W }{| | V ||_{\ MIT V }}. \]
If the \ (\ mit w \) space is a private space, \ (\ scr l (\ mit V; \ mit W) \) is also a private space.
If \ (\ mit W = \ r \), \ ({\ color {red} \ scr l (\ mit V; \ mathbf \ r )}\) is the dual space of \ (\ mit V \). It is often remembered as \ ({\ color {red} \ mit V '}\).
Dual pair (Duality pairingIs called the dual pair between \ (\ mit V \) and \ (\ mit V,
\ [\ Begin {Align *} <\ cdot \, \, \ cdot> &: \ mit V '\ times \ mit V \ longrightarrow \ r \ & <L, v> \ longmapsto L (V ). \ end {Align *} \]
Various convergence Definitions
Strong Convergence: The sequence \ (\ {v_n \} \) in the linear space \ (\ mit V \) is weak in the \ (V \, \ In \ mit V \) refers to the convergence by the norm, namely \ (| v_n-v | \ rightarrow0 (n \ rightarrow \ infin ). \)
Weak Convergence: The sequence \ (\ {v_n \} \) in the linear space \ (\ mit V \) is weak in the \ (V \, \ In \ mit V \) it refers to the convergence of \ (L (v_n) \) to \ (L (V), \) for any \ (L \ In \ mit V ),\) that is, \ (| L (v_n)-L (v) | \ rightarrow0 (n \ rightarrow \ infin ). \)
* Weak Convergence: The sequence \ (\ {l_n \} \) in the dual space \ (\ mit V' \) is weak in \ (L \, \ In \ mit V' \) refers to any one \ (V \ In \ mit V \), all
? \ (| L_n (V)-L (v) | \ rightarrow 0 (n \ rightarrow \ infin ).\)
- \ (Strong convergence of {\ color {red} \ mit V \ rightarrow weak convergence .}\)
- \ (Weak convergence in {\ color {red} \ mit V' \ rightarrow * Weak Convergence .}\)
\ (\ Mit L ^ {p} (\ Omega) \) Space \ (\ Omega _ {open} \ subset \ r ^ {d }\), \ (d \ Ge 1 \), and \ (\ Omega \) isLebesgueMeasurable.
- \ [\ Begin {Align *} \ mit L ^ {p} &: = \ left \ {v \, \, \ big | \ int _ {\ Omega }\, \ left | V (x) \ right | \, ^ p \, DX \ Leq \ infin \ right \}, \, 1 \ Leq p \, <\ infin, \ mit L ^ {\ infin} &: = \ sup \ left \ {| V (x) | \ big | \,\, X \ In \ Omega \ right \} <\ infin. \ end {Align *} \]
? Its norm is
\ [\ Begin {Align *} | v | _ {\ mit L ^ {P }}&:=\ left (\ int _ {\ Omega }\, \ left | V (x) \ right | \, ^ p \, DX \ right) ^ {1/p}, \, 1 \ Leq p \, <\ infin, \ | v | _ {\ mit L ^ {\ infin }&:=\ sup \ left \ {| V (x) | \ big | \,\, X \ In \ Omega \ right \} \ end {Align *} \]
Numerical Solution of partial differential equations-learning Summary