SymPDE concepts and their mathematical meaning

SymPDE concepts and their mathematical meaning#

sympde notation	Mathematical notion
`Domain('Omega', dim=n)`	an abstract domain \(\Omega \subset \mathbb{R}^n\)
`Domain('Omega', interiors, boundaries, connectivity)`	an abstract domain \(\Omega \subset \mathbb{R}^n\), represented by its interiors, boundaries and connectivity
`Interval('I', coordinate=x)`	an abstract (interior) interval \(I \subset \mathbb{R}\), with the coordinate \(x\)
`Union(A, B, C)`	union of domains \(A \cup B \cup C\)
`ProductDomain(A, B)`	product of two domains \(A \times B\)
`Indicator(A)`	Indicator/characteristic function on \(A\), i.e. \(\mathbf{1}\_A\)
`Line('Omega')`	an abstract line segment, \(\Omega \subset \mathbb{R}\), having one interval as interior domain
`Square('Omega')`	an abstract square, \(\Omega \subset \mathbb{R}^2\), having one interior domain as product of two intervals
`Cube('Omega')`	an abstract cube, \(\Omega \subset \mathbb{R}^3\), having one interior domain as product of three intervals
`NormalVector('n')`	Normal vector \(\mathbf{n}\)
`TangentVector('t')`	Tangent vector \(\mathbf{t}\)
`e = ElementDomain(Omega)`	an element of an abstract domain \(e \in \mathcal{T}(\Omega)\) or \(e = d\Omega\), where \(\mathcal{T}(\Omega)\) is a given tessellation of \(\Omega\)
`DomainArea(Omega)`	Area of an abstract domain \(\Omega\)
`ElementArea(Omega)`	Area of an abstract element of a domain \(\Omega\)
`Area(A)`	Area of an expression of topological domain notions

sympde notation	Mathematical notion
`Mapping('F', n)`	a mapping functor \(\mathbf{F}\_{\Omega} := \left( \mathbf{F}, \Omega \right): \Omega \rightarrow \mathbb{R}^n\)
`F[i]`	\(i^{th}\) physical coordinate from \(\{x,y,z\}\)
`F.jacobian`	the jacobian matrix \(\mathcal{D}\_\mathbf{F}\)
`F.det_jacobian`	determinant of the jacobian matrix, \(J_\mathbf{F}:= \mathrm{det} ~\mathcal{D}\_\mathbf{F}\)
`F.covariant`	the covariant matrix \(\left( \mathcal{D}\_\mathbf{F} \right)^{-T}\)
`F.contravariant`	the contravariant matrix \(\frac{1}{J_\mathbf{F}} \mathcal{D}\_\mathbf{F}\)
`F.hessian`	the hessian matrix \(\mathcal{H}\_\mathbf{F}\)
`Covariant(F, v)`	action of the covariant matrix of \(\mathbf{F}\) on \(\mathbf{v}\), i.e. \(\left( \mathcal{D}\_\mathbf{F} \right)^{-T} \mathbf{v}\)
`Contravariant(F, v)`	action of the contravariant matrix of \(\mathbf{F}\) on \(\mathbf{v}\), i.e. \(\frac{1}{J_\mathbf{F}} \mathcal{D}\_\mathbf{F} \mathbf{v}\)

sympde notation	Mathematical notion
`ScalarFunctionSpace('V', Omega)`	scalar function space \(\mathcal{V}\)
`VectorFunctionSpace('W', Omega)`	vector function space \(\mathbf{\mathcal{W}}\)
`ProductSpace(V, W)` or `V*W`	product of spaces, i.e. \(\mathcal{V} \times \mathcal{W}\)
`element_of_space(V, 'v')`	scalar function \(v \in \mathcal{V}\)
`element_of_space(W, 'w')`	vector function \(\mathbf{w} \in \mathcal{W}\)
`element_of_space(V*W, ['v', 'w'])`	\(\left(v,\mathbf{w}\right) \in \mathcal{V} \times \mathcal{W}\)

sympde notation	Mathematical notion
`ScalarFunctionSpace('V0', Omega, kind='H1')`	scalar function space \(\mathcal{V}\_0 \subseteq H^1(\Omega)\)
`VectorFunctionSpace('V1', Omega, kind='Hcurl')`	vector function space \(\mathcal{V}\_1 \subseteq H(\mbox{curl}, \Omega)\)
`VectorFunctionSpace('V2', Omega, kind='Hdiv')`	vector function space \(\mathcal{V}\_2 \subseteq H(\mbox{div}, \Omega)\)
`ScalarFunctionSpace('V3', Omega, kind='L2')`	scalar function space \(\mathcal{V}\_3 \subseteq L^2(\Omega)\)

sympde notation	Mathematical notion
`P_V := Projector(V)`	Projector onto the function space \(\mathcal{V}\)
`P_V(expr)`	Projection of the expression `expr` onto the function space \(\mathcal{V}\)
`Pi_V := Projector(V, 'commuting')`	Commuting projector onto the typed function space \(\mathcal{V}\)
`Pi_V(expr)`	Commuting projection of the expression `expr` onto the typed function space \(\mathcal{V}\)

sympde notation	Mathematical notion
`ScalarFunction(V, 'u')`	\(u \in \mathcal{V}\)
`VectorFunction(W, 'w')`	\(\mathbf{w} \in \mathcal{W}\)
`Constant('mu', real=True)`	constant number \(\mu \in \mathbb{R}\)
`int`	integer number
`float`	floating-point number

boldface font is used for vector functions/expressions

sympde notation	Mathematical notion
`dx(u)`, `dy(u)` or `dz(u)`	\(\partial_x u\), \(\partial_y u\) or \(\partial_z u\)
`grad(u)`	\(\nabla u\) or \(\nabla \mathbf{u}\)
`div(u)`	\(\nabla \cdot \mathbf{u}\)
`curl(u)`	\(\nabla \times \mathbf{u}\)
`rot(u)`	2D rotational \(\nabla \times u\)
`convect(a, u)`	\(\left( \mathbf{a} \cdot \nabla \right) \mathbf{u}\)
`trace(u)`	trace \(\gamma(u)\)
`Dn(u)`	normal derivative \(\partial_{\mathbf{n}} u\)
`D(u)`	Strain tensor \(\frac{1}{2}\left(\nabla \mathbf{u} + \nabla \mathbf{u}^T \right) \)
`laplace(u)`	Laplace \(\Delta u\)
`hessian(u)`	Hessian \(H(u)\)
`bracket(u,v)`	Poisson Bracket \([u,v]\)

sympde notation	Mathematical notion
`jump(u)`	jump of \(u\), i.e. \([u]\)
`avg(u)`	average of \(u\), i.e. \(\langle u \rangle\)
`conv(K,u)`	convolution of \(u\) with a kernel \(K\), i.e. \(K * u\)

sympde notation	Mathematical notion
`DomainIntegral(f)`	integral over a domain, i.e. \((f, \Omega) \mapsto \int_{\Omega} f ~d\Omega\)
`BoundaryIntegral(f)`	integral over a boundary, i.e. \((f, \Gamma) \mapsto \int_{\Gamma} f ~d\partial\Omega\)
`PathIntegral(F)`	integral over an oriented path, i.e. \((\mathbf{F}, C) \mapsto \int_{C} \mathbf{F} \cdot d\mathbf{s}\)
`SurfaceIntegral(F)`	integral over an oriented surface, i.e. \((\mathbf{F}, S) \mapsto \int_{S} \mathbf{F} \cdot d\mathbf{S}\)