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An H1-density rate for P0 single-layer Galerkin BEM

Statement

Assume the Axiom of Choice and Countable Choice. Under the mesh and coercivity hypotheses of P0 Galerkin BEM for the single layer is well posed and quasioptimal, suppose each mesh triangle is contained in one planar face and the exact density λ∈L2(Γ) is continuously differentiable on each face with ∇Γλ∈L2(Γ). Then the P0 Galerkin solution obeys ∥λ−λh∥H−1/2(Γ)≤Ch3/2∥∇Γλ∥L2(Γ), where C is independent of h for a shape-regular mesh family. More generally, the same estimate holds whenever λ∣T∈H1(T) on every mesh triangle, with the gradient norm taken elementwise. No agreement of density traces across mesh edges is required.

Facts & Assumptions

Given: The Axiom of Choice and Countable Choice, the exact and P0 Galerkin solutions, the shape-regular mesh family, and λ∣T∈H1(T) on every triangle; the facewise version assumes that every triangle lies in a single face.

[A1]

Assume the Axiom of Choice and Countable Choice (The Axiom of Choice, The Axiom of Countable Choice (ACω)); these supply the foundational hypotheses of both analytical suppliers.

[F1]

The Galerkin error is bounded by a mesh-independent constant times best approximation in H−1/2 (P0 Galerkin BEM for the single layer is well posed and quasioptimal).

[F2]

Element averages satisfy ∥λ−Πhλ∥H−1/2≤Ch3/2∥∇Γλ∥L2 for densities whose restrictions to every mesh triangle belong to H1(T), using the elementwise gradient norm (P0 averages approximate smooth densities in H negative one half).

Proof

1.1A1F1givenchoose

In [F1], choose the admissible comparison density μh=Πhλ∈Xh. Then the best-approximation infimum is at most ∥λ−Πhλ∥H−1/2.

2.1F2step 1.1givenalgebra∎

In the facewise version, face alignment and the stated regularity give λ∣T∈H1(T) for every triangle; otherwise this is assumed directly. Apply [F2] to the right-hand side of step 1.1 and absorb the mesh-independent quasi-optimality factor into C. This gives the displayed h3/2 estimate.

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