Lowest-order equivalent nonstandard finite element methods for biharmonic plates

The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the $C^0$ interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side $F\in H^{-2}(\Omega)$ replaced by $F\circ (JI_{\rm M}) $ and then are quasi-optimal in their respective discrete norms. The smoother $JI_{\rm M}$ is defined for a piecewise smooth input function by a (generalized) Morley interpolation $I_{\rm M}$ followed by a companion operator $J$. An abstract framework for the error analysis in the energy, weaker and piecewise Sobolev norms for the schemes is outlined and applied to the biharmonic equation. Three errors are also equivalent in some particular discrete norm from [Carstensen, Gallistl, Nataraj: Comparison results of nonstandard $P_2$ finite element methods for the biharmonic problem, ESAIM Math. Model. Numer. Anal. (2015)] without data oscillations. This paper extends the work [Veeser, Zanotti: Quasi-optimal nonconforming methods for symmetric elliptic problems, SIAM J. Numer. Anal. 56 (2018)] to the discontinuous Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms.

For a general right-hand side ∈ −2 (Ω), the standard right-hand side ( ℎ ) remains undefined for nonstandard finite element methods. A postprocessing procedure in [6] enables to introduce a new 0 IP method for right-hand sides in −2 (Ω). In [41][42][43], the discrete test functions are transformed into conforming functions ( is called smoother in those works) before applying the load functional and quasi-optimal energy norm estimates The papers [41][42][43] discuss minimal conditions on a smoother for each problem, while this paper presents one smoother M for all schemes; the best-approximation for the dGFEM is a new result. The smoother M also allows a post-processing with a priori error estimates in weaker and piecewise Sobolev norms. Table 1 summarizes the notation of spaces, bilinear forms, and an operator for the four second-order methods for the biharmonic problem detailed in Subsection 3.

Contributions
The main contributions of this paper are (a) the design and analysis of a generalized Morley interpolation operator M for piecewise smooth functions in 2 (T ), (b) the design of modified schemes for Morley FEM, dGFEM, 0 IP method, and a weakly over-penalized symmetric interior penalty (WOPSIP) method for the biharmonic problem for data in ∈ −2 (Ω), (c) an abstract framework for the best-approximation property and weaker (piecewise) Sobolev norm estimates, (d) a priori error estimates in (piecewise) Sobolev norms for the lowest-order nonstandard finite element methods for biharmonic plates, (e) an extension of the results of [13] to an equivalence

Prologue
This section characterizes the best-approximation property of a class of non-conforming finite element methods. The biharmonic problem is put in an abstract framework in real Hilbert spaces and and a bounded bilinear form : × → R satisfying an inf-sup condition. Given a right-hand side ∈ * , the exact problem seeks ∈ with The discrete problem is put in an analog framework with finite-dimensional real Hilbert spaces ℎ and ℎ and a bilinear form ℎ : ℎ × ℎ → R. The discrete space ℎ (resp. ℎ ) is not a subspace of (resp. ) in general, but and ℎ (resp. and ℎ ) belong to one common bigger vector space that gives rise to the sum = + ℎ (resp. = + ℎ ). It is not supposed that this is a direct sum, so the intersection ∩ ℎ (resp. ∩ ℎ ) may be non-trivial. We suppose that and are real Hilbert spaces with (complete) subspaces , ℎ and , ℎ . The linear and bounded map ∈ ( ℎ ; ) links the right-hand side ∈ * of the exact problem to the right-hand side ℎ := • ∈ * ℎ of the discrete problem. The map is called smoother in [41][42][43] because it maps a (possibly) discontinuous function ℎ ∈ ℎ to a smooth function ℎ in applications. The resulting discrete problem seeks ℎ ∈ ℎ with ℎ ( ℎ , •) = ( •) in ℎ . (2.1) We also suppose that the exact and discrete problems are well-posed and this means in particular that dim ℎ = dim ℎ < ∞ and that the bounded bilinear forms and ℎ satisfy inf-sup conditions with positive constants and ℎ and are non-degenerate such that the associated linear and bounded operators ∈ ( ; * ) and ℎ ∈ ( ℎ ; * ℎ ) are bĳective; the associated linear operators are defined by := ( , •) ∈ * for all ∈ and by ℎ ℎ := ℎ ( ℎ , •) ∈ * ℎ for all ℎ ∈ ℎ . The general discussion in [41][42][43], leads to an optimal smoothing = Π | ℎ for the orthogonal projection Π ∈ ( ) onto . This is a global operation in general and hence infeasible for practical computations. Notice carefully that all examples in [41][42][43] discuss ∈ ( ℎ ; ) with = for all ∈ ℎ ∩ , abbreviated by = id in ℎ ∩ . (2.2) This paper introduces a smoother M for the discontinuous Galerkin schemes that satisfies (2.2) and is quasi-optimal in the following sense with a constant Λ Q ≥ 0 that is exclusively bounded in terms of the shape regularity of the underlying triangulations.
This diagram also depicts some (linear and bounded) operator : ℎ → that will become a quasi-optimal smoother in the context of the best-approximation property of below. A synonym to the best-approximation property of is to say is quasi-opimal in the following sense. Definition 2.2 (quasi-optimal). The above operator is said to be quasi-optimal if A first characterisation of (QO) has been given in [41] in terms of Lemma 2.2. [41] Under the present notation, (QO) is equivalent to (2.4).
The above lemmas characterize qo and Λ Q by a compactness argument and it remains to control qo and Λ Q in terms of mesh-size independent bounds in applications. This paper designs in Section 4.3 a smoother in the spirit of [41][42][43] based on earlier work in the context of a posteriori error control [9,12,16] and adaptive mesh-refinement [19,[23][24][25]. The outcome is a quasi-optimal smoother ∈ ( ℎ ; ) with a constant Λ P that depends only on the shape regularity of the underlying finite element mesh and for all ℎ ∈ ℎ and all ∈ . (2.5) The proof of the following characterization of best-approximation shall be given in the appendix.
In particular, if (H) holds, then (QO) follows with a constant qo that depends solely on ℎ , Λ H , Λ P , , and . The next theorem presents a key estimate that is crucial for goal-oriented error control and duality arguments for weaker norm estimates. The proof and the dependence of contants are presented in the appendix.The motivation for (QO) is exemplified in Theorem 2.5 below.
Theorem 2.4. Suppose and are quasi-optimal smoothers with (2.3)-(2.3) and suppose (QO). Then the existence of qo > 0 with In particular, if (H) holds, (QO) follows with a constant qo that depends solely on , Λ ′ 2 , Λ P , and Λ Q . The a priori error estimates in weaker Sololev norms (weaker than the energy norm) are a corollary of Theorem 2.4 and the elliptic regularity, the latter is written in an abstract form by the assumption that and are two Hilbert spaces with ⊂ and ⊂ such that (R) ∃ reg > 0 ∀ ∈ * − * ≤ reg * for the solution := − * ∈ ⊂ to (•, ) = ∈ * ⊂ * .

Theorem 2.5 (weak a priori). Under the assumptions of Theorem 2.4, (QO) and (R) imply
Proof. Given − ∈ ⊂ , a corollary of the Hahn-Banach extension theorem leads to some ∈ * ⊂ * with norm * ≤ 1 in * and − = ( − ). The dual solution ∈ to = (•, ) ∈ * satisfies (R) and (QO) leads to for any ℎ ∈ ℎ . This and ≤ reg * ≤ reg conclude the proof. (Ω), = + (Ω) and 1/2 ≤ ≤ 1, = 1 or 2. Typical first-order approximation properties of the discrete finite element spaces result in in terms of the maximal mesh-size ℎ max of the underlying finite element mesh. Remark 2.1 (best-approximation constant). The paper [41] gives a formula for the bestapproximation constant qo for some slightly simpler problem in one Hilbert space. Remark 2.2 (injective smoother). Under the above notation ∈ ( ℎ ; ) is injective if and only if ∈ ( ; ℎ ) is surjective [41]. Then there exists a right-inverse ∈ ( ℎ ; ) to and (H) holds with Λ H = 0 (this follows with the arguments of the proof of Theorem 2.5 for ′ that is in fact a quasi-optimal smoother owing to (10.10).) Consequently, the discrete scheme is equivalent to a conforming Petrov-Galerkin scheme. Remark 2.3 (non injective smoother). In case ∈ ( ℎ ; ) is not injective, the discrete problem may reduced to the range ′ ℎ := R ( ) of and the orthogonal complement ′ ℎ of the kernel of in ℎ . However, the explicit computation of the reduced discrete spaces ′ ℎ and ′ ℎ may be costly and hence this paper outlines a general analysis that allows non-injective quasi-optimal smoothers.  2) below for details) followed by a companion operator from the previous example for the dG FEM. Then dim 2 (T ) = 6|T | is strictly larger than dim M(T ) = |V (Ω)| + |E (Ω)|; whence cannot be injective. The situation for the 0 IP with the discrete space 2 0 (T ) (of the same dimension as M(T )) is more involved and is discussed in more details in Section 8 below.

Continuous model problem
Suppose ∈ := 2 0 (Ω) solves the biharmonic equation Δ 2 = for a given right-hand side ∈ * ≡ −2 (Ω) in a planar bounded Lipschitz domain Ω with polygonal boundary Ω. The weak form of this equation reads with the scalar product ( , ) := ∫ Ω 2 : 2 dx for all , ∈ . It is well known that (3.1) has a unique solution and elliptic regularity [1,3,32,38] holds in the sense that ∈ − (Ω) implies ∈ ∩ 4− (Ω) for all with 2 − reg ≤ ≤ 2 with the index of elliptic regularity reg > 0. The lowest-order nonconforming finite element schemes suggest a linear convergence rate in the energy norm for a solution ∈ (Ω) at most for all ≥ 3. Therefore := min{1, reg } is fixed throughout this paper and exclusively depends on Ω. The regularity is frequently employed in the following formulation.  It is true that pure Dirichlet boundary conditions in the model example lead to > 1/2 and then allow for a control of the traces 2 in the jump terms. This paper circumvents this argument and all the results hold for ≥ 0. The new discrete analysis is therefore much more flexible and allows for generalizations of the model problem e.g. for mixed and boundary conditions of less smoothness.

Nonconforming discretisation
Throughout the rest of this article, the following notations are adopted. Let T denote a shape regular triangulation of the polygonal Lipschitz domain into compact triangles. Associate its piecewise constant mesh-size ℎ T ∈ 0 (T ) with ℎ := ℎ T | := diam( ) ≈ | | 1/2 in any triangle ∈ T of area | | and its maximal mesh-size ℎ max := max ℎ T . Let V (resp. V (Ω) or V ( Ω)) denote the set of all (resp. interior or boundary) vertices in T . Let E (resp. E (Ω) or E ( Ω)) denote the set of all (resp. interior or boundary) edges. The length of an edge is denoted by ℎ . Let Π denote the 2 (Ω) orthogonal projection onto the piecewise polynomials Let the Hilbert space 1 (T ) ≡  The semi-scalar product pw is defined by the piecewise differential operator 2 pw and pw ( pw , pw ) := ∈T ∫ 2 pw : 2 pw dx for all pw , pw ∈ 2 (T ).   and ∈ E (Ω). It satisfies (a) the integral mean property of the Hessian, 2 pw M = Π 0 2 ,
A reformulation of Lemma 3.1.a is the best-approximation property pw ( − M , 2 ) = 0 for all ∈ and all 2 ∈ 2 (T ). (In fact = 0.25745784465 from [11] is independent of the shape of the triangle .)

Companion operator and best-approximation for the Morley FEM
A conforming finite-dimensional subspace of 2 0 (Ω) is provided by the Hsieh-Clough-Tocher ( ) FEM [26,Chap. 6]. For any ∈ T , let K ( ) := { : ∈ E ( )} denote the triangulation of into three sub-triangles := conv{ , mid( )} with edges ∈ E ( ) and common vertex mid( ) depicted in Figure 1.b. Then, The degrees of freedom in a triangle ∈ T are the nodal values ( ) and its derivative ∇ ( ) of the function ∈ (T ) at any vertex ∈ V ( ) and the values / (mid( )) of the normal derivatives at the midpoint mid( ) of any edge ∈ E ( ).
This is critical in eigenvalue analysis or problems with low-order terms; for e.g. in [19,23]. The 2 orthogonality in Lemma 3.2.e also allows a direct proof of Theorem 3.3 that circumvents the a posteriori error analysis of the consistency term as part of the medius analysis [34]. Notice that the proof of the best-approximation of Theorem 3.4 for the modified scheme does not require the 2 orthogonality in Lemma 3.2.e. Remark 3.4 (minimal assumptions on the smoother). The series of papers [41]- [43] addresses the question on the minimal assumptions on the smoother (partly as a right inverse only). This paper utilizes a smoother with the properties of Lemma 3.2.a-d.
The point in the subsequent example is that the smoother M may be more costly than averaging in other examples but it is at almost no extra costs for the case of point forces, which are of practical importance in civil engineering.

Equivalent norms
The Hilbert space 2 The homogeneous boundary conditions in 2 0 (Ω) are included in the the jump contributions at the boundary with jump partner zero owing to the homogeneous boundary conditions in (3.1).
The discontinuous Galerkin schemes of [2,28] are associated with a another family of norms • dG depending on the two positive parameters 1 , 2 > 0 in the semi-norm scalar product for all pw , pw ∈ 2 (T ). The DG norm • dG is the square root of for all pw ∈ 2 (T ). It depends on the parameters 1 , 2 > 0 and so do all constants in the sequel; in particular those suppressed in the abbreviations and ≈. The conditions on the ellipticity of the scheme in Lemma 7.1 below will assert that 1 and 2 are sufficiently large. The analysis of this paper assumes this and simplifies the notation 1 ≈ 1 ≈ 2 .
The (possibly discontinuous) piecewise affine interpolation 1 ∈ 1 (T ) of pw ∈ 2 (T ) is defined by nodal interpolation 1 | ( ) = pw | ( ) at the three vertices ∈ V ( ) in each triangle ∈ T . It is well known from standard finite element interpolation [4,6,26] that the error : for each triangle ∈ T with explicit constants [14] that exclusively depend on the maximal angle in the triangulation. The nodal interpolation implies is an affine function along the edge ∈ E, an inverse estimate shows with a triangle inequality in the last step for = pw − 1 . (The constant ℎ /6 in the first inequality of (4.6) stems from the eigenvalues ℎ /2 and ℎ /6 of the 2 × 2 mass matrix of piecewise linear functions in 1D.) This implies an estimate for the first term of the definition of ℎ ( pw ): A typical contribution ( ⨏ pw / ds) 2 for the second term (in the definition of ℎ ) is controlled with a Cauchy inequality by ℎ −1 pw / 2 2 ( ) . This results in shared by the two triangles ± ∈ T plus trace inequalities on ± show with (4.5) in the end. The omission of − in the above arguments for an edge ∈ E ( Ω) on the boundary provide (4.7) with ( ) = + . This and the finite overlap show The upper bound in the latter estimate is pw 2 dG up to the weights 1 Recall the piecewise affine interpolation 1 ∈ 1 (T ) of pw ∈ 2 (T ) and := pw − 1 ∈ 2 (T ) with (4.5) from the previous part of the proof. Standard trace inequalities as in (4.7) for the first term (and an analog for the second term This and triangle inequalities result in pw pw 2 ( ( )) . The constant factor 1/2 in the upper bound of the first subsequent inequality (displayed as 2 in the lower bound) stems from the eigenvalues ℎ /2 and ℎ /6 of the 2 × 2 mass matrix of piecewise linear functions in 1D, with the nodal interpolation property 1 | ( ) = pw | ( ) for ∈ V ( ), ∈ T , in the last step. The jump [ 1 / ] is constant along the edge and so with a triangle inequality in the last step. A Cauchy inequality ∇ pw 1 ( ) ≤ ℎ 1/2 ∇ pw 2 ( ) and a trace inequality show (as above in (4.8)) that The combination of all aforementioned estimates reads The sum of all those estimates over ∈ E plus ||| pw ||| 2 pw leads to an estimate with the lower bound pw 2 dG up to the weights 1 ≈ 1 ≈ 2 . The finite overlap of the edge-patches ( ( ) : ∈ E) shows that the resulting upper bound is pw ℎ .
Proof of the upper bound. The proof of the asserted inequality starts with triangle inequalities for the jumps of pw ∈ 2 (T ) and the shape regularity for ℎ ≈ ℎ for ∈ E ( ). This and a Cauchy inequality A one-dimensional trace inequality (with a factor 1 that follows from 1D integration) with standard trace inequalities on for pw and ∇ pw in the last step. The right-hand side The remaining details are omitted for brevity.

Interpolation errors
The interpolation error estimates are summarised in one theorem.

Theorem 4.2 (interpolation). Any pw ∈ 2 (T ) and its Morley interpolation
Proof of (a). The first step reduces the analysis to piecewise quadratic functions by the piecewise Morley interpolation loc M . Definition 3.1 shows ∫ ∇( loc M pw − pw )| ds = 0 for an edge ∈ E ( ) of the triangle and therefore 2 ( loc M pw )| = Π 0 2 pw | a.e. in ∈ T . Notice that the piecewise defined Morley interpolation 2 := loc M pw ∈ 2 (T ) is discontinuous (and shares none of the compatibility or boundary conditions) in general. The interpolation error estimates of Lemma 3.
This and a triangle inequality show that it remains to prove that M := M pw satisfies for the jump terms localised to a neighbourhood Ω( ) of ∈ T as follows. The neighbourhood Ω( ) is the interior of the union ∪{ ∈ T : dist( , ) = 0} of ∈ T plus one layer of triangles in T around. Then is the contribution from and its neighbourhood Ω( ) to the full jump term ℎ ( pw ) 2 with the spider E ( ) := { ∈ E : ∈ V ( )} of edges with one end-point ∈ V ( ). The second step reduces the analysis to piecewise quadratic functions. The first obervation is that the averaging of the degrees of freedom in the definition of M merely employs the data of 2 = loc M pw in the sense that M = M pw = M 2 . This explains why ℎ ( pw , ) = ℎ ( 2 , ) in the asserted estimate (4.9). The second observation is that the left-hand side of (4.9) involves the polyonomial ( 2 − M )| ∈ 2 ( ) that allows for inverse estimates The overall conclusion is that it suffices to prove, for all 2 ∈ 2 (T ), that In fact, (4.10) and the aforementioned arguments lead to a localised form of the assertion.
The sum over all ∈ T and the bounded overlap of (Ω( ) : ∈ T ) then conclude the proof of the theorem. The third step reduces the proof of (4.10) to six coefficients. The six degrees of freedom on a triangle ∈ T are the three point evaluations at the three vertices ∈ V ( ) and the three integral means of the normal derivatives ⨏ • ds along the three edges ∈ E ( ). The six dual basis functions for ∈ V ( ) and for ∈ E ( ) in 2 ( ) are defined by the duality relations ( ) = 0 = ⨏ ds and ( ) = 1 = ⨏ ds for all ∈ V ( ) and ∈ E ( ), while ( ) = 0 = ⨏ ds for all vertices ≠ ∈ V ( ) and edges ≠ ∈ E ( ). Those functions are known and given explicitly (e.g., in [12] in the context of a short implementation of the Morley FEM in 30 lines of Matlab) with a scaling (which is generally understood and follows from the explicit formulas) . A triangle inequality in this sequence 1 , . . . , shows that − ≤ =1 | +1 − | (even with an omitted factor 1/2). It follows | 1 |, . . . , | | ≤ =1 | +1 − | and so, for a triangle ∈ T ( ) in the notation of (4.11), follows (with a Cauchy inequality in R in the end). This is suboptimal and the best constant in a squared version of this argument is contained in [23,Appendix C]. Observe that 1 is bounded from above by the shape regularity of the triangulation T .  The fitfh step finishes the proof. Recall that the coefficients ( ) for ∈ V ( ) and ( ) for any ∈ E ( ) in (4.11) satisfy (4.12)-(4.13). The resulting estimate reads with the shape regularity ℎ ≈ ℎ for ∈ E ( ) and ∈ V ( ) in the end. This concludes the proof of (4.10) and thus that of (a) as outlined at the end of the second step.
Proof of (b). Given any M ∈ M(T ), part (a) shows that the first term 1 in the equivalence

Approximation errors
The subsequent theorem discusses the approximation properties of • M for piecewise smooth and piecewise quadratic functions. It is formulated in terms of • ℎ ≈ • dG and the norm equivalence implies an (undisplayed) analog for • dG as well.
This and the trivial estimate ℎ 2( −1) The sum over all those contributions over ∈ T proves with Theorem 4.3.c in the last step. This concludes the proof of (d) for 1 < < 2. The assertion (d) is included in Theorem 4.3.c for = 0, 1, 2. The remaining case 0 < < 1 is similar to the above analysis with ( ) ≤ 2 ( ) 1− 2 ( )

Discretisation
Suppose that ℎ ⊂ 2 (T ) is the finite-dimensional trial and test space of an abstract (discontinuous Galerkin) scheme with a bilinear form that is coercive and continuous with respect to some norm • ℎ in 2 (T ) in the sense that, for all ℎ , ℎ ∈ ℎ , ℎ 2 ℎ ≤ ℎ ( ℎ , ℎ ) and ℎ ( ℎ , ℎ ) ≤

First glance at the analysis
This subsection motivates the abstract conditions and emphasises the relevance of the discrete consistency condition (dcc) that leads to the best-approximation in terms of ||| − M ||| pw = min 2 ∈ 2 ( ) ||| − 2 ||| pw from (3.7). The test function ℎ := ℎ M − ℎ ∈ ℎ ⊂ 2 (T ) is the discrete approximation of the error − ℎ with M : 2 (T ) → M(T ) from Definition 3.2 and a transfer operator ℎ : M(T ) → ℎ from Subsection 5.3 below. For the dGFEM of Section 7 and the WOPSIP scheme of Section 10, ℎ is the identity 1 and otherwise it is controlled nicely (cf. (5.11) below for details) [13]. So we may neglect the difference 1 − ℎ for the sake of this first look at the analysis and suppose ℎ = 1. The key identity from the continuous problem (3.1) and the discrete one (5.3) reads The stability of the scheme for the three bilinear forms that define the class of problems in (5.12) displayed in Table 1.
The definition of ℎ in (5.6) leads to Since is a the right-inverse of M , (3. This in combination with (5.5) leads in (5.7) to The second term in the right-hand side of (5.  The examples of this paper concern the discrete norm from (4.1)-(4.2) and then the estimates of this subsection follow for piecewise quadratic discrete spaces.

Sufficient conditions for best-approximation
The bilinear forms ℎ , pw , ℎ , ℎ : The key assumption in abstract form is the discrete consistency condition with a constant 0 < Λ dc < ∞: All functions M ∈ M(T ), ℎ ∈ ℎ , and all , ∈ satisfy (This is a straightforward generalization of (5.4) from Subsection 5.2.) Assume that ℎ : Remark 5.3. The Morley FEM is included in the (non-symmetric) abstract framework of Theorem 5.1 and leads to a sub-optimal best-approximation constant qo = 1 + Λ 0 .
The error analysis of a post-processing dates back at least to [6] with a design of an enrichment operator for C 0 IP functions replaced here by the smoother M . For ∈ − (Ω) with 2 − ≤ ≤ 2 (and ∈ 4− (Ω) from elliptic regularity), Theorem 5.1 verifies
Proof. The discrete consistency condition 6 Weaker and piecewise Sobolev norm error estimates

Lemma 6.2 (Key identity). It holds
Proof. Let ≡ M ℎ ∈ substitute the test function in (3.1). This and the test function (3.5) and (3.9)) and elementary algebra conclude the proof.

Elementary bounds Lemma 6.3. Each of the following terms (a)
Proof.

Proof of Theorem 6.1
Given 2 − ≤ ≤ 2, there exists a constant 0 < int ( ) < ∞ (which exclusively depends on the shape regularity of T and ) such that the solution ∈ of the dual problem in Section 6.2 satisfies (with Lemma 3.1.c) that Proof of (a). Recall − (Ω) = ( − , ) from Subsection 6.2 and its formula in Lemma 6.2. Lemma 6.3 applies to the first four terms and Lemma 6.4 to the remaining three. The resulting estimate reads . This and ||| − M ||| pw ≤ int ( ) reg ( )ℎ 2− max from (6.13) prove Theorem 6.1.a.
Since ( 2 − M 2 )| ± resp. (1 − ) M | ± is a polynomial of degree at most 2 resp. 3 in the triangle ± , the discrete trace inequalities hold for a constant 7 ≈ 1 that solely depends on the shape regularity of ± (and so on the shape regularity of T ). This leads to for any interior edge ∈ E (Ω) with the reduced edge-patch ( ). The same estimate follows for a boundary edge ∈ E ( Ω) (the proof omits − , − , and some factor 1/2 above). Since the reduced edge-patches ( ( ) : ∈ E) have no overlap, the sum of all the above estimates of ( ) in (7.6) and Cauchy inequalities prove Recall from Theorem 4.2.a. (with (1 − Π 0 ) 2 pw 2 = 0 for 2 ∈ 2 (T )) that For the right-hand side ∈ −2 (Ω), the modified 0 IP method is based on the continuous Lagrange 2 finite element space ℎ := 2 0 (T ) := 2 (T ) ∩ 1 0 (Ω) and penalty terms along edges. The scheme is a modification of the dGFEM in Section 7 but with trial and test functions restricted to 2 0 (T ) := 2 (T ) ∩ 1 0 (Ω). The norm • IP is • dG with restriction to 2 0 (T ) and excludes one of the penalty parameters of the modified dGFEM. Given IP > 0, the bilinear forms [6,17]   Remark 8.1. A 0 IP discrete scheme is analysed in [6] for a general ∈ −2 (Ω). The consistency of the scheme allows a best approximation [6, Lemma 8] (since ℎ subset 2− (Ω) in the pure Dirichlet problem for 0 IP). For ∈ −2 (Ω), a modifed scheme and error estimates for the post-processed solution are derived in [6, (4.17) and Theorem 4].
Overview of the proof of Theorem 8.1. The proof follows the lines of that of Theorem 7.2 and partly from the analysis provided there. The bilinear forms in the 0 IP are exactly the respective bilinear forms of the dGFEM when restricted to the subspace 2 0 (T ) + (T ). With the single exception of (5.11), all the estimates in (5.9)-(5.16) and (6.1) for Θ = 1 follow for ℎ = 2 0 (T ) in the 0 IP from the respective properties verified in Section 7 for ℎ = 2 (T ) in the dGFEM. The remaining detail is the analysis of the operator

Comparison
The paper [13] has established equivalence of discrete solutions to Morley FEM, 0 IP and dGFEM up to oscillations for ∈ 2 (Ω) and for the original schemes with ℎ ≡ . The subsequent theorem establishes the three modified schemes with ℎ = • without extra oscillation terms. Throughout this section, the norm · ℎ is defined in (4.1)-(4.2).
The equivalence constants ≈ depend on shape regularity and on the stabilisation parameters dG , IP ≈ 1.

Modified WOPSIP Method
The weakly over-penalized symmetric interior penalty (WOPSIP) scheme [5] is a penalty method with the stabilisation term for piecewise smooth functions pw , pw ∈ 2 (T ). This semi-norm scalar product P (•, •) is an analog to that one behind the jump ℎ from (4.2) with different powers of the mesh-size. It follows as in Theorem 4.1 that P ( pw , pw ) := pw ( pw , pw ) + P ( pw , pw ) for all pw , pw ∈ 2 (T ) (10.2) defines a scalar product and so • P := P (•, •) 1/2 is a norm in 2 (T ). Consequently, there exists a unique solution P ∈ ℎ := 2 (T ) to P ( P , 2 ) = ( M 2 ) for all 2 ∈ 2 (T ). (10. 3) The increased condition number in the over-penalization of the jumps by the negative powers of the mesh-size in (10.1) can be compensated by some preconditioner [5, p 218f] and the entire WOPSIP linear algebra with (10.3) becomes intrinsically parallel. The constant Λ P exclusively depends on the shape regularity of T , while 8 ( ) depends on the shape regularity of T and on .
The subsequent lemma specifies the constant Λ P in the best-approximation estimate.
Proof of Lemma 10.2. The analysis of |||ℎ −1 T ( 2 − M 2 )||| pw returns to the proof of Theorem 4.2 that eventually provides (4.11) for one fixed triangle ∈ T with its neighourhood Ω( ) for any 2 ∈ 2 (T ). The substitution of 2 by ℎ −1 2 (with a fixed scaling factor ℎ ) in (4.11) after a standard inverse estimate shows, for all 2 ∈ 2 (T ), that The shape regularity of T implies that all edge-sizes in the sub-triangulation T (Ω( )) that covers the neighbouhood Ω( ) (of and one layer of triangles around ) are equivalent to ℎ . Hence ℎ (ℎ −1 2 , ) is equivalent to the respective contributions in P ( 2 , 2 ): The combination of this estimate with the previous one and the sum over all those estimates lead to