This section is devoted for a model scenario from phase transition problems  with some solution u that is smooth outside some one-dimensional interface Γ. Suppose some (possibly non-unique) minimiser u of the continuous problem (2.3) satisfies for some finite union Γ of (n - 1) dimensional Lipschitz surfaces in . Since Ω has a Lipschitz boundary, this implies Lipschitz continuity of u on Ω. We refer to  for sufficient conditions for and conclude that the remaining assumption is the essential hypothesis expected in many interface problems. Let be the (unique) minimiser of the discrete stabilised problem (2.5). In the following, also Γ = ∅ is permitted to extend previous results  for highly regular minimisers.
The following theorem leads to a priori convergence rates for the interface model problem. Thereby it recovers the results of  for problems with piecewise smooth exact solution.
We will abbreviate the set of all triangles that are touched by Γ as , its cardinality as , its union as with volume |ΩΓ,ℓ| and its complement as .
Provided β > 0, it holds
In the case of uniform mesh refinements we may expect and |ΩΓ,ℓ| ≈ H
and Theorem 5.1 simplifies to
= (id - I
= (id - I
)u, a Young inequality, (3.3) and (, Theorem 3.8) yield
Theorem 4.4.4 in  shows and
Let be the patch of a side , and set and . Note that [Du]
= 0 for . Then
The first sum can be estimated as in the proof of (, Lemma 3.2), the second sum with
The observation concludes the proof.□
Together with Theorem 5.1, the subsequent result implies strong convergence of the gradients in the model interface problem as H
Under the aforementioned conditions on the (possibly non-unique) exact minimiser, the error e
= u - u
of the discrete solutionof (2.5) satisfies
The basic idea of gradient control is the generalisation of the interpolation estimate H1(Ω) = [L2(Ω), H2(Ω)]1/2 for a reduced domain Ω∖Γ; refer to [24, 25] for a detailed analysis of interpolation spaces. Let w
be the boundary value interpolation of (id - I
as described in (, Prop. 4.1), such that w
satisfies the inequalities in (4.2). A piecewise integration by parts shows, for , that
where nΓ is a unit normal vector of the interface Γ. The Lipschitz continuity of u implies |[Du]ΓnΓ| ≲ 1. This and the trace inequality on Γ lead to
The case is contained in (, Theorem 4.4). The piecewise Laplacian of u is bounded in L2(Ω) and so (with the generic constant hidden in the notation C ≈ 1)
The elementwise trace inequality (, Theorem 1.6.6, p. 39) for an n-dimensional simplex T and one of its sides F, and , 1 ≤ q < ∞, reads
The term and the stabilisation ∥ |u
are already analysed in the Estimate on C in the proof of (, Theorem 4.4). This results in
The preceding estimates plus the absorbtion of lead to
The triangle inequality applied to v = e
and some careful elementary analysis to absorb eventually lead to
The inequalities (4.2), Poincaré and Friedrichs inequalities on sides and removal of higher-order terms in H
conclude the proof.□
The following theorem is an improved a posteriori estimate based on Theorems 4.1 and 5.3.
Recall, the definitions e
:= u - u
:= σ - σ
for σ := DW(Du) and σ
), and the definition of Λ
from Section ‘A posteriori error estimates’. Set
Provided β>0, it holds
The generic constants in Theorem 5.4 depend on problem-specific data such as the shapes of Ω and Γ as well as the generic constant ϰ of Theorem 4.1.
Theorem 5.4 holds verbatim in Example 3.3 and in the modified two-well problem of Subsection ‘Modified two-well benchmark’, where β = 0.
The assertion of Theorem 5.4 holds for any discrete u
∈ uD,ℓ + V
which may approximate the discrete unique exact solution of (2.5). This allows the inexact SOLVE via an iterative procedure.
Proof of Theorem 5.4
as in the proof of Theorem 5.3. Then Theorem 4.1 with q = 2 and (4.2) imply
Theorem 5.3 provides an estimate of the semi-norm . A Young inequality shows . The absorbtion of then proves the first assertion. The second assertion is an immediate consequence of the first one, Theorem 5.3 and several algebraic transformations.