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Briefly, what are the different types of basis used in FEM and why is the nodal basis so popular and advantageous in the finite element context?

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People use all sorts of bases in practice. For example, people use orthonormal bases in DG methods to ensure that the mass matrix in time stepping schemes is diagonal. People also use hierarchical bases when doing $p$ adaptivity because it makes the construction of constraints at faces where different polynomial degrees come together trivial. For higher order methods, people also use other constructions to minimize the condition number of the matrix.

In other words, there is a wide variety of bases that are in actual use. We just happen to start teaching the FEM with nodal bases because it's so easy to understand and because they're sufficient for most cases.

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The nodal Lagrange bases are nice because they interpolate the functions at the knots: $$\phi_i(x_j) = \delta_{ij}$$ This means that you can read and plot solutions by looking only at the coefficients $u_j$ in the representation:$$u_h(x) = \sum_j u_j\phi_j(x)$$ That's much nicer than having to evaluate the sum at every point you care about knowing $u_h$.

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In Engineering nodal bases are a good starting point for solid mechanics problems because the principle of virtual work for the discretised system $\mathbf{K} \mathbf{u} = \mathbf{f}$ reads $$ \delta \mathbf{u}^T \mathbf{K} \mathbf{u} = \delta \mathbf{u}^T \mathbf{f} $$ showing that $\mathbf{K} \mathbf{u}$ are indeed (equivalent) nodal internal forces, and $\mathbf{f}$ are (equivalent) nodal external forces. For other bases, although the principle of virtual work is still valid, $\mathbf{u}$ are not nodal displacements, and therefore the meaning of $\mathbf{f}$ is much more abstract.

Early FEM developments were mostly driven by engineering applications, and the intuition of point forces applied at nodes was very important for the method diffusion.

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There are a variety of different bases in FEM, but most involve basis functions which are associated with topological entities, like vertices, edges, faces, and element interiors. This makes it possible to enforce various types of continuity by ensuring that degrees of freedom for such functions match at shared vertices/edges/faces.

These basis functions can also be defined in a hierarchal fashion (define 1D functions, blend those into 2D functions, blend 2D functions into 3D, etc). Bases defined this way can be used to expose sparsity or guarantee other mathematical properties, though their construction is more involved.

Nodal bases are simpler way to define such functions, so long as an appropriate number of nodes are placed on the vertices, edges, faces, and interior of an element. Continuity can be enforced by ensuring that two nodal values are identical at shared vertices/edges/faces. Additionally, if such nodes are co-located at quadrature points, this can be exploited for efficient time-stepping and mass-matrix assembly on quadrilateral and hexahedral elements (this is at the root of the Spectral Element Method).

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Ideally, one would like to have an orthonormal basis with respect to the $L_2$ or energy inner product, or both (e.g., the eigenmodes of the operator at hand), because many matrices would be diagonal and thus very easy to invert. But that's hard to get (solving eigenproblems is a nonlinear procedure). Lagrange bases is provides sparse matrices, which in some sense is the next best choice to diagonal, and it very easy to build; so they provide the best of both worlds. But some more sophisticated choices, such as hierarchical, wavelet bases, Bernstein polynomials, are indeed possible and work better in special applications.

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