Variational methods and finite elements

For −Δu=f with zero Dirichlet data, consider J(v)=one half the integral of |∇v|² minus the integral of fv over a suitable bounded domain. Perturb v by εw with w in H1_0 and differentiate at ε=0. Stationarity gives ∫∇u·∇w=∫fw, exactly the weak problem. For this symmetric coercive case the stationary point is the unique minimum, not an arbitrary saddle. Nonlinear energies, nonconvex coefficients, or different boundary conditions need their own analysis rather than inheriting that conclusion automatically.

The abstract setting is a bounded bilinear form a on a Hilbert space V and a bounded linear functional ℓ. If a(v,v) is at least α times the V norm squared for some α>0, Lax–Milgram gives a unique solution of a(u,v)=ℓ(v) for all tests. For the zero-Dirichlet Poisson problem, Poincaré's inequality connects the gradient norm to the full H1 norm. On a pure Neumann space constants destroy coercivity unless one passes to mean-zero functions or a quotient. Boundary choices therefore influence both the theorem and the matrix nullspace.

Let V_h be a conforming subspace of H1_0 made of continuous piecewise-linear functions on a mesh. Choose basis functions φ_j and write u_h=ΣU_jφ_j. Testing against each basis function produces K_ij=∫∇φ_j·∇φ_i and F_i=∫fφ_i, giving KU=F. The coefficients U are nodal values for the usual linear nodal basis, but that is a property of this basis rather than a universal definition of a finite element coefficient. Mesh geometry, element order, boundary treatment, and quadrature all affect the assembled system.

The discrete solution satisfies the weak equation only for tests in V_h. Subtracting the continuous and discrete equations gives Galerkin orthogonality a(u−u_h,v_h)=0. In the symmetric positive case this makes u_h the energy-orthogonal projection and gives best approximation in the energy norm. Céa's lemma gives an analogous constant-factor estimate for bounded coercive forms. Neither result promises a particular power of mesh width without regularity of the exact solution and suitable mesh approximation properties. Singular corners can limit regularity even when the data look smooth.

Solving KU=F accurately removes algebraic error for the chosen discrete problem, not necessarily the error relative to the continuum solution. Mesh refinement addresses approximation error, while accurate quadrature addresses integration error. A continuous piecewise-linear function has zero second derivative inside each element but derivative jumps at element interfaces, so its pointwise strong residual is not the whole story. The weak form includes the effect of those jumps through integration. Residual-based error estimators must account for element contributions, interface jumps, and the boundary model.

The function lab plots only y=ax², its tangent at x, and its signed integral from zero to x. Set a=−1 and move x to compare the smooth quadratic shape with a linear tangent; observe where a local straight-line description differs from the curve. This is an analogy for local approximation, not a finite element mesh, an assembled stiffness matrix, or an error estimator. The one-node calculation below supplies the actual finite element construction and checks it against the known quadratic solution.