Strang error bound for a parabolic isoparametric P1 family

Given: The Axiom of Choice and Countable Choice; the parabolic exact and approximate geometry maps on the reference square; a shape-regular matching mesh with $h\le 1$ and $J_h\ge 1\mathrm{/}2$; bounded, uniformly coercive forms and bounded exact and discrete loads; the stated exact and discrete variational solutions; and, when a lift is present, the stated decomposition of the load difference.

$H^k=W^{k,2}$ with the derivative-sum Sobolev norm, and $H_0^1=W_0^{1,2}$ is the norm closure of $C_c^{\mathrm{\infty }}$ in $H^1$. (The notation $H^k$ and the reserved zero-boundary symbol, Integer-order Sobolev spaces and their norms, Zero-boundary Sobolev space as a norm closure)

A matching affine mesh is a finite face-to-face triangulation by nondegenerate cells; shape regularity gives the cellwise scaling hypotheses used by the P1 interpolation estimate. (Matching affine simplicial meshes and shape regularity, Shape regularity controls affine simplex geometry)

$e_0$ and $e_1$ denote the uniform value and derivative geometry gaps. (Curved-domain geometry approximation errors)

For the parabolic geometry family, $e_0\le C{h}^2$, $e_1\le Ch$, and $J_h\ge 1\mathrm{/}2$ for sufficiently small $h$. (P1 approximation of a parabolic shear map)

The pullback of the physical Laplace form has tensor $JD{\mathrm{\Phi }}^{-1}D{\mathrm{\Phi }}^{-T}$, and the tensor difference obeys $\mathrm{\parallel }B-B_h{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\le C{e}_1$. (Pullback gradients and induced surface metrics, Derivative geometry error controls the Laplace tensor defect)

On each affine triangle, the identity-map P1 interpolant of an $H^2$ function has $H^1$ error at most $C{h}_K\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2(K)}$; the constants are uniform over the shape-regular mesh family. (P1 interpolation scaling on uniformly curved cells)

The nodal P1 interpolant of $u\in H^2\cap H_0^1$ is well-defined, continuous across the mesh, zero on boundary edges, and belongs to $V_h$. (The nodal P1 interpolant preserves homogeneous boundary values)

The first Strang estimate bounds the error by best approximation, the form defect, and the load defect divided by the uniform discrete coercivity constant. (First Strang estimate for conforming perturbed Galerkin problems)

The variational-crime framework defines ${\delta }_{a,h}(v_h)={\sup }_{w_h\mathrm{\ne }0}\mathrm{\mid }a(v_h,w_h)-a_h(v_h,w_h)\mathrm{\mid }\mathrm{/}\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_{H^1}$ and ${\delta }_{F,h}={\sup }_{w_h\mathrm{\ne }0}\mathrm{\mid }F(w_h)-F_h(w_h)\mathrm{\mid }\mathrm{/}\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_{H^1}$. (Variational-crime settings and consistency defects)

For the common $C^2$ physical lift, the pulled-back residual satisfies $\mathrm{\parallel }{\mathrm{\ell }}_{\mathrm{\partial },h}{\mathrm{\parallel }}_{V^{\ast }}\le C_G(e_0+e_1)$. (Geometry error bounds a pulled-back boundary-lift residual)

With the standard $H^1$ inner product, $H_0^1(\widehat{\mathrm{\Omega }})$ is a closed Hilbert subspace of $H^1(\widehat{\mathrm{\Omega }})$. (The Sobolev space $H^1$ is a Hilbert space)

Countable Choice is assumed through the first-Strang, geometry and interpolation interfaces. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))

The Axiom of Choice is assumed for the automatic nodal-boundedness clause of the interpolation supplier and the local conformity result. (The Axiom of Choice)