Weak formulations and Galerkin orthogonality

Let the domain be bounded and Lipschitz, impose homogeneous Dirichlet data on its whole boundary, and set V equal to H¹₀ of that domain. Take a scalar diffusion coefficient kappa that is essentially bounded between two positive constants. The gradient seminorm is a norm on V by Poincaré’s inequality. An integrable source is not automatically a bounded functional in every dimension; assume directly that the load belongs to the dual space V′, or use an L² source, which suffices on this bounded domain.

Multiply the differential equation by a zero-trace test function and integrate by parts, initially for smooth data and then by density. The boundary term vanishes because of the test trace, yielding the weak problem below. The bilinear form is continuous with constant kappa-max and coercive with constant kappa-min in the gradient norm. Lax–Milgram therefore supplies a unique weak solution. No global H² regularity is needed for this existence statement, and a weak solution need not have classical second derivatives everywhere.

Choose a finite-dimensional subspace V_h contained in V and seek u_h there using the same bilinear form and load functional. Every discrete test is then admissible in the continuous problem. Subtracting the two equations gives a residual that vanishes on V_h. This is Galerkin orthogonality. It says nothing about tests outside that subspace, and it is not a claim that the strong PDE residual vanishes pointwise inside every element or at every node.

For a symmetric positive bilinear form, orthogonality is orthogonality in its energy inner product. For a nonsymmetric coercive form the displayed zero residual still holds, but one cannot automatically use a Pythagorean identity or interpret it as Euclidean perpendicularity. A stiffness matrix is obtained by testing basis expansions; its entries encode this variational form. Changing the form by approximate integration or choosing nonconforming test functions changes the subtraction argument and must be analyzed explicitly.

A solved algebraic system makes the weak residual vanish against its chosen basis. The remaining discretization error comes from omitted directions in the infinite-dimensional space. Conversely, an inexact linear solve leaves an algebraic residual even within the discrete space, so one must separate solver error from approximation error. Refining a mesh increases the available tests but does not justify silently dropping either component when assessing an actual computed solution.

The clean homogeneous formulation also explains why boundary conditions matter to the space. A constant function has zero gradient energy, so a pure Neumann problem needs a compatibility condition and normalization rather than the same coercivity assertion on unrestricted H¹. Nonzero Dirichlet values lead to an affine trial set and a zero-trace test space, handled later by lifting. The first invariant to check in any implementation is therefore the continuous and discrete space pair, followed by which exact forms are actually assembled.