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Some Notations

Here mathematical expressions and corresponding FreeFem++ commands are explained.

Generalities#

  • [\delta_{ij}] Kronecker delta (0 if i\neq j, 1 if i=j for integers i,j)
  • [\forall] for all
  • [\exists] there exists
  • [i.e.] that is
  • [PDE] partial differential equation (with boundary conditions)
  • [\emptyset] the empty set
  • [\N] the set of integers (a\in \N\Leftrightarrow int a), int means long int inside FreeFem++
  • [\R] the set of real numbers (a\in \R\Leftrightarrow real a), double inside FreeFem++
  • [\C] the set of complex numbers (a\in \C\Leftrightarrow complex a), complex<double>
  • [\R^d] d-dimensional Euclidean space

Sets, Mappings, Matrices, Vectors#

Let E,\, F,\, G be three sets and A the subset of E.

  • [\{x\in E|\; P\}] the subset of E consisting of the elements possessing the property P
  • [E\cup F] the set of elements belonging to E or F
  • [E\cap F] the set of elements belonging to E and F
  • [E\setminus A] the set \{x\in E|\; x\not\in A\}
  • [E+F] E\cup F with E\cap F=\emptyset
  • [E\times F] the Cartesian product of E and F
  • [E^n] the n-th power of E (E^2=E\times E, E^n=E\times E^{n-1})
  • [f:\; E\to F] the mapping form E into F, i.e., E\ni x\mapsto f(x)\in F
  • [I_E or I] the identity mapping in E,i.e., I(x)=x\quad \forall x\in E
  • [f\circ g] for f:\; F\to G and g:\; E\to F, E\ni x\mapsto (f\circ g)(x)=f(g(x))\in G (see Elementary function)
  • [f|_A] the restriction of f:\; E\to F to the subset A of E
  • [\{a_k\}] column vector with components a_k
  • [(a_k)] row vector with components a_k
  • [(a_{k})^T] denotes the transpose of a matrix (a_{k}), and is \{a_{k}\}
  • [\{a_{ij}\}] matrix with components a_{ij}, and (a_{ij})^T=(a_{ji})

Numbers#

For two real numbers a,b

  • [a,b] is the interval \{x\in \R|\; a\le x\le b\}
  • ]a,b] is the interval \{x\in \R|\; a< x\le b\}
  • [a,b[ is the interval \{x\in \R|\; a\le x< b\}
  • ]a,b[ is the interval \{x\in \R|\; a< x< b\}

Differential Calculus#

  • [\p f/\p x] the partial derivative of f:\R^d\to \R with respect to x (dx(f))
  • [\nabla f] the gradient of f:\Omega\to \R,i.e., \nabla f=(\p f/\p x,\, \p f/\p y)
  • [div\mathbf{f} or \nabla.\mathbf{f}] the divergence of \mathbf{f}:\Omega\to \R^d, i.e., div\mathbf{f}=\p f_1/\p x+\p f_2/\p y
  • [\Delta f] the Laplacian of f:\; \Omega\to \R, i.e., \Delta f=\p^2f/\p x^2+\p^2 f/\p y^2

Meshes#

  • [\Omega] usually denotes a domain on which PDE is defined
  • [\Gamma] denotes the boundary of \Omega,i.e., \Gamma=\p\Omega (keyword border, see Border)
  • [\mathcal{T}_h] the triangulation of \Omega, i.e., the set of triangles T_k, where h stands for mesh size (keyword mesh, buildmesh, see Mesh Generation)
  • [n_t] the number of triangles in \mathcal{T}_h (get by Th.nt)
  • [\Omega_h] denotes the approximated domain \Omega_h=\cup_{k=1}^{n_t}T_k of \Omega. If \Omega is polygonal domain, then it will be \Omega=\Omega_h
  • [\Gamma_h] the boundary of \Omega_h
  • [n_v] the number of vertices in \mathcal{T}_h (get by Th.nv)
  • [n_{be}] the number of boundary element in \mathcal{T}_h (get by Th.nbe)
  • [|\Omega_h|] the measure (area or volume) in \mathcal{T}_h (get by Th.measure)
  • [|\partial \Omega_h|] the measure of the border (length or area) in \mathcal{T}_h (get by Th.bordermeasure)
  • [h_{min}] the minimum edge size of \mathcal{T}_h (get by Th.hmin)
  • [h_{max}] the maximum edge size of \mathcal{T}_h (get by Th.hmax)
  • [[q^iq^j]] the segment connecting q^i and q^j
  • [q^{k_1},q^{k_2},q^{k_3}] the vertices of a triangle T_k with anti-clock direction (get the coordinate of q^{k_j} by (Th[k-1][j-1].x, Th[k-1][j-1].y)
  • [I_{\Omega}] the set \{i\in \N|\; q^i\not\in \Gamma_h\}

Finite Element Spaces#

  • [L^2(\Omega)] the set \displaystyle{\left\{w(x,y)\left|\; \int_{\Omega}|w(x,y)|^2\d x\d y<\infty\right.\right\}}
  • [H^1(\Omega)] the set \displaystyle{\left\{w\in L^2(\Omega)\left|\; \int_{\Omega}\left(|\p w/\p x|^2+|\p w/\p y|^2\right)\d x\d y <\infty\right.\right\}}
  • [H^m(\Omega)] the set \displaystyle{\left\{w\in L^2(\Omega)\left|\; \int_{\Omega}\frac{\p^{|\alpha|} w}{\p x^{\alpha_1}\p y^{\alpha_2}}\in L^2(\Omega)\quad\forall \alpha=(\alpha_1,\alpha_2)\in \N^2,\, |\alpha|=\alpha_1+\alpha_2\right.\right\}}
  • [H^1_0(\Omega)] the set \left\{w\in H^1(\Omega)\left|\; u=0\quad \textrm{on }\Gamma\right.\right\} * [L^2(\Omega)^2] denotes L^2(\Omega)\times L^2(\Omega), and also H^1(\Omega)^2=H^1(\Omega)\times H^1(\Omega)
  • [V_h] denotes the finite element space created by fespace Vh(Th, *) in FreeFem++ (see Finite Elements for *)
  • [\Pi_h f] the projection of the function f into V_h (func f=x^2*y^3; Vh v = f;} means v = Pi_h (f) * [\{v\}] for FE-function v in V_h means the column vector (v_1,\cdots,v_M)^T if v=v_1\phi_1+\cdots+v_M\phi_M, which is shown by fespace Vh(Th, P2); Vh v; cout << v[] << endl;