From Basis to Shape Functions

The discrete formulation requires choosing a basis $\{\phi_1, \ldots, \phi_N\}$ for the space $V_h$ and assembling the stiffness matrix $K_{ij} = a(\phi_j, \phi_i)$ and load vector $f_i = \ell(\phi_i)$ . In practice these global basis functions are not constructed directly; they emerge from simpler, locally defined functions called shape functions.

Partition $\bar{\Omega}$ into $M$ non-overlapping elements $T_1, \ldots, T_M$ whose interiors are disjoint and whose union covers $\bar{\Omega}$ . On each element $T_e$ a small set of shape functions is defined, each supported entirely within $T_e$ . The global space $V_h$ is then assembled from these local pieces — either by identifying shape functions on adjacent elements along shared boundaries (as in the standard continuous Galerkin method) or by keeping them independent (as in the discontinuous Galerkin method). Either way, each global basis function $\phi_i$ has support on at most a small number of elements.

Since $a(\cdot, \cdot)$ is bilinear, the stiffness matrix entry decomposes as

K_{ij} = a(\phi_j, \phi_i) = \sum_{e=1}^{M} a\big|_{T_e}(\phi_j, \phi_i),

where $a\big|_{T_e}$ denotes the restriction of $a$ to element $T_e$ . The contribution from $T_e$ is zero unless the supports of both $\phi_i$ and $\phi_j$ overlap with $T_e$ , which can only happen for nearby basis functions. This is the origin of the sparsity of the stiffness matrix. Assembly therefore proceeds element by element: for each $T_e$ one computes a small element stiffness matrix and scatters its entries into the global $K$ . The load vector is assembled the same way, each element contributing a small element load vector to $\mathbf{f}$ .