the discontinuous Galerkin method
/ gah-LYAIR-kin /
The finite element method and the finite volume method each have a great strength the other lacks. Finite elements give high-order accuracy by using rich polynomials, and they handle complex geometry; finite volumes guarantee exact conservation and handle shocks gracefully via numerical fluxes. The discontinuous Galerkin method is the hybrid that takes both: it is a finite element method that, like finite volumes, allows the solution to JUMP between elements and couples neighbours through numerical fluxes. It is often described as the bridge between FEM and FVM.
Here is the construction. Like FEM, you put a polynomial of degree p inside each element and use a Galerkin weak form. But unlike standard FEM, you do NOT require continuity across element boundaries — each element carries its own independent polynomial, and the solution can be discontinuous at element interfaces, exactly like the cell averages of FVM (which are the special case p=0). To couple the disconnected elements, you do what finite volumes do: integrate by parts within each element, and on each shared face replace the (now two-valued) flux by a single NUMERICAL FLUX computed from both sides. So DG combines a high-order polynomial reconstruction inside elements with a flux-based, conservative coupling between them. The mass matrix is block-diagonal (each element is independent), which makes explicit time-stepping very efficient and highly parallel.
Discontinuous Galerkin has become a leading method for high-order computational fluid dynamics, electromagnetics, and convection-dominated problems, precisely because it marries high-order accuracy with local conservation and shock capturing on unstructured meshes. The honest costs are real. DG carries MANY more degrees of freedom than continuous FEM — every element duplicates the unknowns on its shared faces, so a DG mesh has several times more unknowns than a continuous one of the same order. It needs the same nonlinear limiting machinery as finite volumes to stay non-oscillatory at shocks (and limiting high-order DG cleanly is genuinely hard). And explicit DG inherits the same CFL time-step restriction that gets tighter as p rises. It is powerful but not a free upgrade — you pay in degrees of freedom and implementation complexity for the accuracy and flexibility.
With degree p=0 (one constant per element), DG reduces exactly to the finite volume method — each cell holds one average and neighbours couple through a numerical flux. Raising to p=1, 2, 3 puts a linear, quadratic, or cubic polynomial in each cell, lifting the accuracy to O(h^{p+1}) on smooth solutions while keeping the same flux-based, conservative coupling between cells.
FVM is DG at p=0; raising p gives high-order accuracy with the same conservative flux coupling.
DG buys high-order accuracy plus local conservation, but at a real cost: it duplicates unknowns on shared faces (far more degrees of freedom than continuous FEM), still needs limiters at shocks, and its explicit time step tightens as the polynomial degree grows. It is a powerful trade, not a free one.