PDEresidualer

PDEresidualer are the discrepancies that arise when a candidate function is substituted into a partial differential equation (PDE). For a PDE written as L[u] = s on a domain Ω with boundary conditions B[u] = g on ∂Ω, the residual for an approximate solution û is r(x) = L[û](x) − s(x) in the domain, together with a boundary residual r_B(x) = B[û](x) − g(x) on the boundary. If û were an exact solution, these residuals would be zero; in practice they are generally nonzero and quantify how well the PDE is satisfied by the approximation.

Residuals can be categorized by form. A strong residual is evaluated pointwise and is common in collocation methods and physics-informed neural networks (PINNs), where r(x) is enforced at selected points. A weak residual arises in variational or weak formulations, where the residual is tested against a set of functions φ and integrated, leading to the equations used in finite element and spectral methods.

Applications of PDE residuals include assessing solution quality, guiding error estimation, and driving adaptive refinement. The norm of the residual (for example, the L2 norm) often serves as an objective function in residual-minimization approaches and helps compare different numerical approximations. In inverse problems, residual minimization is a standard strategy to identify model parameters consistent with observed data.

Common examples illustrate the concept. For the heat equation u_t − κΔu = 0, the strong residual is r = u_t − κΔu; for Poisson’s equation −Δu = f, the residual is r = −Δu − f (or Δu + f, depending on sign convention). PDE residuals provide a unifying view of how well a proposed solution satisfies a PDE across methods.