Lower-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 0 interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side ∈ −2 (Ω) replaced by ◦ ( M) and then are quasi-optimal in their respective discrete norms. The smoother M is defined for a piecewise smooth input function by a (generalized) Morley interpolation M followed by a companion operator . 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 %2 finite element methods for the biharmonic problem, ESAIM Math. Model. Numer. Anal. (2015)] without data oscillations. This paper extends the work [Veeser, Zanotti: Quasioptimal 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 [34,35], 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 [33][34][35] 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 2.2 and in Sections 6, 7, and 9.

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 [10] to an equivalence without data oscillations for the modified schemes, (f) the proof of the best approximation for the modified dGFEM that extends [34] and [10,Thm 4.3].

Organization
The remaining parts of this paper are organised as follows. Section 2 presents preliminaries, a nonconforming discretisation, introduces a novel generalized Morley interpolation operator for discontinuous functions and states an abstract best-approximation [14] result for nonconforming discretisation when data ∈ −2 (Ω). Section 3 proves a crucial equivalence result of two discrete norms for a piecewise 2 (T ) function, and proves approximation properties for the generalized Morley interpolation operator. Section 4 provides an abstract framework for dG methods and the proof of a best-approximation property under a set of general assumptions. Section 5 develops the abstract result for a priori error estimates in weaker and piecewise Sobolev norms. Sections 6 and 7 recall the dG and 0 IP schemes and verify the assumptions of Sections 4 and 5 for the best-approximation result in the energy norm as well as weaker and piecewise Sobolev norms without data oscillations. The paper concludes with the equivalence (1.1) of errors in Section 8 and a proof of quasi-optimality up to penalty for the WOPSIP scheme in Section 9.

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 ( , ) = ( ) for all ∈ (2.1) with the scalar product ( , ) := ∫ Ω 2 : 2 dx for all , ∈ . It is well known that (2.1) has a unique solution and elliptic regularity [1,3,25,31] 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. Example 2.1 (regularity). There exists a constant 0 < ≤ 1 such that ∈ − (Ω) with 2 − ≤ ≤ 2 satisfies ∈ ∩ 4− (Ω) and for some constant reg ( ) < ∞, which depends on Ω and . (The dependence on results from the equivalence of Sobolev norms that may depend on the index in general.) 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.

Interpolation of discontinuous functions
Throughout this article, T is a shape regular triangulation of Ω and M(T ) is given as above.

Definition 2.3 (Morley interpolation II)
. Given any pw ∈ 2 (T ), define M pw := M ∈ M(T ) by the degrees of freedom as follows. For any interior vertex ∈ V (Ω) with set of attached triangles T ( ) that has cardinality |T ( )| ∈ N and any interior edge = + ∩ − ∈ E (Ω) and its mean value operator • (the arithmetic mean of the two traces from the triangles + and − ∈ T along their common edge = + ∩ − ), set  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 ( ). [18,22,33]). There exists a linear map : M(T ) → ( (T ) + 8 (T )) ∩ 2 0 (Ω) and a constant Λ J (that exclusively depends on the shape regularity of T ) such that any M ∈ M(T ) satisfies (a)-(e). (2.10)

Remark 2.4.
The Morley FEM is included in the (non-symmetric) abstract framework of Theorem 4.1 below and leads to a sub-optimal best-approximation constant qo = 1 + Λ 0 . ℎ T ( M − M ) for 0 ≤ ≤ 2. This is critical in eigenvalue analysis or problems with low-order terms; for e.g. in [15,18]. The 2 orthogonality in Lemma 2.2.e also allows a direct proof of Theorem 2.3 that circumvents the a posteriori error analysis of the consistency term as part of the medius analysis [27]. Notice that the proof of the best-approximation of Theorem 2.4 for the modified scheme does not require the 2 orthogonality in Lemma 2.2.e. Remark 2.6 (minimal assumptions on the smoother). The series of papers [33]- [35] 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 2.2.a-d. In fact, it is a key observation that the extra orthogonality in Lemma 2.2.e is not needed to establish the results in this paper with a modified right-hand side ℎ = • . Hence for the practical realization, only the properties of Lemma 2.2.a-d suffice and that is realized by a simple right inverse based on averaging of nodal degrees of freedom [18,35]. However, the extra computational costs for the 2 orthogonality in Lemma 2.2.e. are marginal.
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. 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 (2.1).
The discontinuous Galerkin schemes of [2,21] 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 6.1 below will assert that 1 and 2 are sufficiently large. The analysis of this paper assumes this and simplifies the notation 1 ≈ 1 ≈ 2 .
Proof that • ℎ is a norm. Embedding and trace inequalities guarantee that the jump contributions in ℎ ( pw ) are well defined for any pw ∈ 2 (T ) and the remaining conditions on a norm follow directly from the linearity of the involved trace, jump, and integral operators. So the proof focusses on the definiteness: Suppose that 0 = pw ℎ for some pw ∈ 2 (T ) and conclude first that pw ∈ 1 (T ) is piecewise affine and second that pw ∈ M(T ). But pw ∈ 1 (T ) ∩ M(T ) implies that pw ∈ 1 0 (T ) is continuous at the vertices and then also globally (for it is piecewise affine) with zero boundary conditions. The normal derivatives of pw ∈ 1 0 (T ) ∩ M(T ) are constants along any edge and so do not jump across an edge (from pw ∈ M(T )); they vanish along the boundary Ω. This implies pw ∈ 1 0 (T ) ∩ 2 0 (Ω) ⊂ 1 (Ω) is 1 . In particular, the piecewise constant gradients are globally continuous, whence the piecewise constant vector ∇ pw ∈ 0 (Ω; R 2 ) is a global constant vector. The boundary conditions show first ∇ pw ≡ 0 and then pw ≡ 0 in Ω.
. For any 2 ∈ 2 (T ), the condition ℎ ( 2 ) = 0 is equivalent to 2 ∈ M(T ). (This follows from the definitions of M(T ) and ℎ .) 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 [4,6,19] that the error : for each triangle ∈ T with explicit constants [11] 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 (3.6) 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 ): shared by the two triangles ± ∈ T plus trace inequalities on ± show with (3.5) in the end. The omission of − in the above arguments for an edge ∈ E ( Ω) on the boundary provide (3.7) with ( ) = + . This and the finite overlap show The upper bound in the latter estimate is pw 2 dG up to the weights 1 ≈ 1 ≈ 2 .
Proof of • dG • ℎ . Recall the piecewise affine interpolation 1 ∈ 1 (T ) of pw ∈ 2 (T ) and := pw − 1 ∈ 2 (T ) with (3.5) from the previous part of the proof. Standard trace inequalities as in (3.7) for the first term (and an analog for the second term This and triangle inequalities result in 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 (3.8)) 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 ∇ pw 1 ( ) ≤ ℎ 1/2 ∇ pw 2 ( ) lead to 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 is 2 =0 |ℎ −2 T pw | 2 (T ) as asserted. The remaining details are omitted for brevity.

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

Theorem 3.2 (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 2.
This and a triangle inequality show that it remains to prove that M : 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 (3.9). The second observation is that the left-hand side of (3.9) involves the polyonomial The overall conclusion is that it suffices to prove, for all 2 ∈ 2 (T ), that In fact, (3.10) 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 (3.10) 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 [9] 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 (3.11), 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 [18,Appendix C]. Observe that 1 is bounded from above by the shape regularity of the triangulation T . In the remaining case of a vertex ∈ V ( Ω) on the boundary, M ( ) = 0 and, in The fitfh step finishes the proof. Recall that the coefficients ( ) for ∈ V ( ) and ( ) for any ∈ E ( ) in (3.11) satisfy (3.12)-(3.13). The resulting estimate reads with the shape regularity ℎ ≈ ℎ for ∈ E ( ) and ∈ V ( ) in the end. This concludes the proof of (3.10) 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. Theorem 3.3 (approximation). Any pw ∈ 2 (T ) and 2 ∈ 2 (T ) satisfy (a)-(d).
Proof of (b). Adapt the notation of part (a) and recall that Theorem 3. Proof of (c). The assertions (a)-(b) apply to pw := 2 ∈ 2 (T ) and the extra term (1 − Π 0 ) 2 pw pw 2 (Ω) vanishes. The resulting estimates allow for obvious converse inequalities and so prove, for 2 ∈ 2 (T ) and M := M 2 ∈ M(T ), that with Theorem 3.1 in between the two equivalences. A triangle inequality, the estimate (3.14.b), the estimate for (1 − ) M in the proof of ( ) and (3.1) applies to pw := 2 ∈ 2 (T ) and shows 2 =0 |ℎ −2 . This concludes the proof of (c). with the subadditivity ( + ) ≤ + for , ≥ 0 and 0 < = − 1 < 1 (e.g. from the concavity of ↦ → for non-negative ) in the last step. An elementary estimate is followed by the Young inequality This and the trivial estimate The sum over all those contributions over ∈ T proves with Theorem 3.3.c in the last step. This concludes the proof of (d) for 1 < < 2. The assertion (d) is included in Theorem 3.3.c for = 0, 1, 2. The remaining case 0 < < 1 is similar to the above analysis with ( ) ≤ 2 ( ) 1− 2 ( ) 1 ( ) replacing (3.16) and analogous arguments; hence further details are omitted.

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 ,

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 for ||| − M ||| pw = min 2 ∈ 2 ( ) ||| − 2 ||| pw from (2.7). 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 2.3 and a transfer operator ℎ : M(T ) → ℎ from Subsection 4.3 below. For the dGFEM of Section 6 and the WOPSIP scheme of Section 9, ℎ is the identity 1 and otherwise it is controlled nicely (cf. (4.10) below for details) [10]. 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 (2.1) and the discrete one (4.3) reads The stabillity of the scheme for the three bilinear forms that define the class of problems in (4.11) displayed in Table 1.
Since is a the right-inverse of M , (2.5) implies pw ( M , M ℎ − M ℎ ) = 0. This and elementary algebra show The combination with (4.5) leads to and so to The second term in the right-hand side of (4.7) is equal to pw ( M − , M ℎ − M ℎ ) and the stabilisation term is equal to ℎ ( M − , ℎ ). They are controlled by ||| − M ||| pw ℎ ℎ . The bilinear form ℎ enjoys the miraculous property ℎ ( M , M ℎ ) = 0 for the discontinuous Galerkin schemes of this paper. The remaining term on the righthand side of (4.7) is pw ( M , ℎ − M ℎ ) + ℎ ( M , ℎ − M ℎ ) and in fact controlled by the dcc (4.4). The proof of dcc in Section 7 is one key argument in this paper. The examples of this paper concern the discrete norm from (3.1)-(3.2) and then the estimates of this subsection follow for piecewise quadratic discrete spaces.  1)-(3.2)). Suppose that the discrete norm • ℎ is defined by (3.1)-(3.2) and ℎ ⊆ 2 (T ). Then (4.2) and (4.8)-(4.9) follow.

Lemma 4.2 (Key identity). It holds
Proof. The test function := M ℎ ∈ in (2.1) and the test function ℎ : (2.5) shows that this is equal to

This and the definition
Proof. For ℎ ∈ ℎ and M ∈ M(T ), the Cauchy inequality plus (4.2.a) show This and the boundedness of ℎ in (4.14) result in The inequality (4.10) with M = M and = reads The combination of the last two displayed inequalities concludes the proof.
The combination of the resulting inequalities concludes the proof.

Proof of best-approximation in Theorem 4.1.a. The discrete ellipticity (4.1) is followed by Lemma 4.2 with terms controlled in Lemmas 4.3-4.5. This leads (after a division by
On the other hand, The combination of the two displayed estimates reads

Assumptions and result
This subsection presents one further condition sufficient for a lower-order a priori error estimate for the discrete problem (4.3) beyond the hypotheses of Subsections 4.1-4.4: The dual discrete consistency with a constant 0 ≤ Λ ddc < ∞ asserts that any ℎ ∈ ℎ , M ∈ M(T ), and any , ∈ satisfy Since ℎ ∈ ℎ may not belong to (Ω), 2 − ≤ ≤ 2 in general, the post-processing M ℎ ∈ arises in the duality argument with 0 < ≤ 1 from Example 2.1.

Lemma 5.2 (Key identity). It holds
Proof. Let ≡ M ℎ ∈ substitute the test function in (2.1). This and the test function
Overview of the proof of Theorem 7.1. The proof follows the lines of that of Theorem 6.2 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 (4.10), all the estimates in (4.8)-(4.15) and (5.1) for Θ = 1 follow for ℎ = 2 0 (T ) in the 0 IP from the respective properties verified in Section 6 for ℎ = 2 (T ) in the dGFEM. The remaining detail is the analysis of the operator  Proof of (4.10). This is included in [10, Lemma 3.2f] in a slightly different notation. In the notation of this paper, Lemma 3.2 of [10] shows for any M ∈ M(T ). Lemma 3.3 of [10] controls the upper bound of (7.5) by the a posteriori terms ∈E ℎ 2 pw M 2 2 ( ) . The latter is efficient, i.e., ||| M − ||| pw for any ∈ 2 0 (Ω). This leads to (4.10). Theorem 3.3 allows for an alternative proof that departs at

Comparison
The paper [10] 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 (3.1)-(3.2).
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 (3.2) with different powers of the mesh-size. It follows as in Theorem 3.1 that P ( pw , pw ) := pw ( pw , pw ) + P ( pw , pw ) for all pw , pw ∈ 2 (T ) (9.2) 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 ). (9. 3) The increased condition number in the over-penalization of the jumps by the negative powers of the mesh-size in (9.1) can be compensated by some preconditioner [5, p 218f] and the entire WOPSIP linear algebra with (9.3) becomes intrinsically parallel.
The constant Λ P exclusively depends on the shape regularity of T , while 9 ( ) depends on the shape regularity of T and on .