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Piecewise $P_2$ finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation
Rui Ma, Yang Xing, Shuo Zhang
TL;DR
The paper addresses the open problem of constructing a lowest-degree P2 discretization for the biharmonic equation on rectangular meshes in arbitrary dimensions. It develops such a weakly continuous space and states an error theorem for shape-regular rectangular meshes when the solution is in H3(Ω).
Problem
A theoretically lowest-degree P2 construction for the biharmonic equation on rectangular meshes in arbitrary dimensions had remained open.
Source: §1 · p. 2Method
The paper constructs a piecewise P2 space with weak continuity of vertex sums and averaged normal derivatives, and develops local patch bases and a quadratic-reproducing interpolation operator.
Source: §1 · p. 2Results
The stated error theorem applies on shape-regular rectangular meshes when u∈H3(Ω).
Source: §2.2 · p. 5Takeaways & Limitations
The construction provides a nonconforming quadratic spline space with compact patch-supported basis functions on rectangular meshes in arbitrary dimensions.
Source: §1 · p. 2The numerical experiment takes u∈H2_0(Ω) and sufficiently smooth.
Source: §2.3 · p. 5
Abstract
from arXiv · showhide
This paper develops a piecewise $P_2$ finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation. The discrete space imposes continuity of vertex sums on $(d-2)$-faces and of averaged normal derivatives across interior $(d-1)$-faces. A local algebraic characterization of $P_2$ is first established through the unisolvence of an enriched polynomial space and the resulting compatibility conditions. For each interior element, a $3^d$-element patch is then introduced, and the associated homogeneous patch space is shown to be one-dimensional. The corresponding patch functions, extended by zero, form a basis of the global homogeneous discrete space. A local quadratic interpolation operator is further constructed in arbitrary dimensions. Its approximation properties, together with a consistency estimate derived from the weak continuity conditions, lead to the optimal broken $H^2$ error estimate for the biharmonic problem.
1. Introduction
Nonconforming methods relax global H2 conformity through weak continuity, and the paper motivates them as a way to reduce polynomial degree and degrees of freedom while retaining optimal approximation. It addresses the open construction of a lowest-degree P2 biharmonic discretization on rectangular meshes in arbitrary dimensions.
1. Introduction
- Nonconforming methods relax global H2 conformity through weak continuity conditions and can reduce polynomial degree and degrees of freedom while retaining optimal approximation properties.The paper places its rectangular-mesh construction against a background where fourth-order nonconforming constructions are less available than those on simplicial meshes.Source: §1 · pp. 1–2
- A piecewise P2 space on rectangular meshes in arbitrary dimensions supplies the previously open lowest-degree construction for the biharmonic problem.Its weak conditions require continuity of vertex sums on (d-2)-faces and averaged normal derivatives across interior (d-1)-faces.Source: §1 · p. 2
- The paper characterizes P2 locally through unisolvence of an enriched polynomial space and compatibility conditions, then proves one-dimensionality of each interior 3^d-element homogeneous patch space.The associated patch functions, extended by zero, form a basis of the global homogeneous space.Source: §1 · p. 2
- A local quadratic-reproducing interpolation operator and approximation properties combine with a consistency estimate derived from the continuity conditions to yield the biharmonic error estimate.The continuity conditions also imply continuity of the face-averaged gradient across interior (d-1)-faces.Source: §1 · p. 2
- The construction uses only P2 polynomials and couples them through weak interelement continuity, yielding a nonconforming quadratic spline with compact patch supports.In two dimensions, its homogeneous space coincides with the reduced quadrilateral Morley space on rectangular meshes.Source: §1 · p. 2
2. Preliminaries and main results
Section 2 sets up the rectangular-mesh biharmonic discretization and states its basis and error results, then reports a three-dimensional convergence experiment.
2.1. Rectangular meshes and finite element spaces.
- The mesh and finite element setup defines rectangular elements, face-average normal derivatives, and vertex-sum functionals; in two dimensions, Vh0 coincides with the homogeneous reduced quadrilateral Morley space.The equivalence is stated for rectangular meshes.Source: §2.1 · pp. 3–4
2.2. Main results.
- For each interior element, the associated patch functions form a basis of the global homogeneous space, whose dimension equals the number of interior elements.The theorem’s basis characterization uses patch functions constructed in Subsection 3.3.Source: §2.2 · p. 5
- The error result assumes u ∈ H3(Ω); its proof controls approximation and consistency terms using separate estimates.The proof applies the second Strang lemma.Source: §2.2 · p. 5
2.3. Numerical experiment in three dimensions.
- The three-dimensional experiment starts from a 4×6×5 rectangular mesh and bisects each element in all coordinate directions at every refinement.The initial mesh is shown in Figure 2.1.Source: §2.3 · p. 5, Figure 2.1 · p. 6
- The smooth-solution experiment shows first-order broken H2 convergence and approximately second-order L2 and broken H1 convergence.The broken H2 rate agrees with the estimate in Theorem 2.2; the results also verify the patch-basis implementation numerically.Source: §2.3 · p. 6
3. Local unisolvence and patch basis
This section establishes local polynomial unisolvence and compatibility, then shows that compatible patch prescriptions uniquely determine local homogeneous functions. Their zero extensions form a basis of the global homogeneous space, with one basis function per interior element.
3.1. Unisolvence for the enriched space P + on a d-dimensional rectangle.
- Theorem 3.6 states that, for d ≥ 2, the enriched space P+ is uniquely determined by N+; the stated dimension of P+ is 2 + 2d.Source: §3.1 · pp. 7–8
3.2. Unisolvence for P2(K) on a d-dimensional rectangle.
- For d ≥2, prescribed (d−2)-face vertex sums and coordinate-derivative averages determine a unique P2(K) function exactly when the compatibility conditions hold.The theorem establishes both necessity and sufficiency of those conditions.Source: §3.2 · pp. 10–11
3.3. Patch basis functions.
- Each interior element has a 3^d-element patch whose homogeneous nonconforming P2 space is one-dimensional; its nonzero basis function extends by zero to Ω.If the mesh has no interior elements, equivalently some N_i ≤2, then Vh0 = {0}.Source: §3.3 · pp. 11, 14
- The argument establishes v|Kα=0 for every α∈{1,2,3}^d and therefore v=0 on M; when the degrees of freedom of v|Kα vanish, Theorem 3.6 gives v|Kα=0.Source: §3.3 · p. 14
- Compatible prescriptions uniquely determine a function in Vh0(MKα), and its zero extension belongs to Vh0.Source: §3.3 · pp. 15, 17, §3.4 · p. 17
3.4. Proof of Theorem 2.1.
- A lexicographic sweep proves spanning by eliminating residuals on lower-corner elements and proves independence using each patch function’s nonzero distinguished vertex-sum.Source: §3.4 · pp. 17–18
- The zero-extended patch functions form a basis of Vh0, with dim Vh0 equal to the number of interior elements.Source: §3.4 · pp. 17–18
4. Interpolation and error estimates
The section constructs a quadratic-preserving interpolant and derives approximation and consistency estimates for the biharmonic problem. Its interpolation design addresses stability, polynomial reproduction, and homogeneous boundary conditions.
4.1. Interpolation operator and optimal approximation.
- The quadratic-preserving interpolation operator generalizes [4] to arbitrary dimension, and its construction combines stable cell averages, a full patch neighborhood, and an interior-center restriction.The boundary-layer defect from restricting centers to the interior is controlled by the vanishing traces of H2_0(Ω) functions.Source: §4.1 · p. 18
4.1.1. Extended mesh and P2 reproduction.
- On each element, the interpolant reproduces P2 by matching its vertex sums and coordinate-derivative averages, which characterize the local quadratic polynomial.The proof applies this local identity on every element of the mesh.Source: §4.1.1 · p. 20
4.1.2. Stable local averaging and the auxiliary interpolant.
- Replacing point-evaluation or second-derivative functionals with L1-stable local averages yields an auxiliary interpolant with polynomial preservation, stability, and local approximation properties.The associated bounds use a constant independent of h and the element.Source: §4.1.1 · p. 20, §4.1.2 · pp. 21–22
4.1.3. Homogeneous interpolant.
- Omitting patch-function contributions centered outside the original interior index set defines an interpolant into Vh0; it differs from the auxiliary interpolant only near the boundary.Each coefficient uses only the 2d+1 cells in its local stencil.Source: §4.1.3 · p. 22
4.1.4. Approximation estimate.
- The approximation argument transfers the local estimate to the homogeneous interpolant, with an additional boundary-layer argument and constants independent of h and v.The boundary-layer control uses the vanishing traces of H2_0(Ω) functions.Source: §4.1.4 · p. 22
4.2. Consistency error estimation.
- Vertex-sum continuity controls averages of tangential derivatives across faces; together with normal-derivative continuity, this supports the consistency estimate for the biharmonic problem.The estimate applies to v ∈ H2_0(Ω) ∩ H3(Ω) with ∆2v ∈ L2(Ω), with a constant independent of h, v, and vh.Source: §4.2 · pp. 23–24
A.1. Proof details for Lemma 4.3.
The appendix verifies the local identities by evaluating the relevant functionals on a monomial basis and applying coordinate-wise weight identities.
A.1. Proof details for Lemma 4.3.
- The proof collects coordinate-direction weight identities as computational ingredients for Lemma 4.3.The identities are applied for each coordinate direction m and index ρ in the stated range.Source: §A.1 · p. 25
- The vertex-sum identity is checked by evaluating the relevant expression on the monomial basis of P2.The cases shown include constants, linear monomials, and quadratic monomials.Source: §A.1 · pp. 25–26
- The coordinate-derivative-average identity is likewise verified on monomials, with separate cases for linear and quadratic terms and symmetry between coordinate directions.The proof concludes that the identity holds by linearity.Source: §A.1 · pp. 26–27
A.2. Proof of Lemma 4.6.
The proof of Lemma 4.6 uses Taylor expansion and coefficient identities to establish its stated relation.
A.2. Proof of Lemma 4.6.
- The proof applies the local identity (4.3) and an exact Taylor expansion at xKα.Source: §A.2 · p. 27
- It isolates the coefficient of the second derivative in the expansion.Source: §A.2 · p. 27
- Summing the coefficient identities over coordinate directions and using the definition of wα,0 completes the displayed calculation.Source: §A.2 · p. 27
A.3. Proof of Lemma 4.10.
The proof of Lemma 4.10 relates the local averaging and point functionals on quadratics, then uses Taylor expansion and trace estimates.
A.3. Proof of Lemma 4.10.
- For p ∈ P2(∆T), Lemmas 4.6 and 4.3 give λKα(p) = rKα(p).Source: §A.3 · p. 28
- Expanding p at the boundary vertex aα yields (4.9), with the pure second-derivative terms canceling in rKα.Source: §A.3 · p. 28
- The proof uses face trace estimates and the vanishing boundary traces of v and its specified derivatives to obtain (4.10).Source: §A.3 · p. 28