Parametrized Differential Equations

Let us first introduce a (suitably regular) physical domain \(\varOmega \in \mathbb R^d\) with boundary \(\partial \varOmega \), where \(d = 1, 2\), or 3 is the spatial dimension. We shall consider only real-valued field variables. However, both scalar-valued (e.g., temperature in a Poisson conduction problems) and vector-valued (e.g., displacement in a linear elasticity problem) field variables \(w:\varOmega \rightarrow \mathbb R^{d_v}\) may be considered: here \(d_v\) denotes the dimension of the field variable; for scalar-valued fields, \(d_v = 1\), while for vector-valued fields, \(d_v = d\). We also introduce (boundary measurable) segments of \(\partial \varOmega \), \(\varGamma ^D_i\), \(1 \le i \le d_v\), over which we will impose Dirichlet boundary conditions on the components of the field variable.