A localized singular behavior in the solution of a partial differential equation that arises at geometric corners or interior angles on the domain boundary, typically manifesting as unbounded gradients or power-law divergence of field quantities.

Principle

The local geometry (interior angle) of a boundary corner changes the regularity class of the PDE solution: eigenmodes tied to the corner produce power-law behaviour r^λ, where the exponent λ depends on the corner angle and boundary conditions, causing reduced convergence rates for standard approximations.

Demonstration

On an L-shaped domain solving Laplace's equation with homogeneous Dirichlet boundary conditions, the reentrant 270° corner produces a solution whose gradient behaves like a negative power of the distance to the corner, causing large errors unless the singularity is resolved or enriched.

Misapplication

Attributing the observed local blow-up solely to a faulty mesh or solver rather than recognizing the geometric origin; removing mesh refinement without addressing the corner singular function will not restore correct convergence.

Consequence

Recognition of a corner singularity forces use of targeted remedies (local mesh refinement, graded meshes, singular function enrichment, or p-adaptivity) to recover accuracy and expected convergence rates in numerical schemes.

Reversal

If the boundary corner is smoothed to a C1 or analytic curve (no geometric corner), the singularity disappears and the solution regains higher regularity consistent with smooth-boundary theory.

Boundary

Applies to boundary-value problems on domains with non-smooth (corner) boundaries for elliptic, and often parabolic or elastostatic, PDEs; does not describe singular behavior caused solely by singular source terms or incompatible boundary data on otherwise smooth boundaries.

Semantic Tension

Often conflated with numerical artifacts (spurious oscillations) or with physical singularities from sources; the corner singularity is a geometrically induced analytic irregularity distinct from discretization-induced noise or from genuine delta-type sources.

Synthesis

Corner singularity is the geometric-driven loss of regularity at domain corners: it is predicted by angle-dependent eigenmodes, observed as power-law blow-up of derivatives, and demands analytic or mesh-based mitigation to restore convergence in numerical solutions.