A recovery based linear finite element method for the Monge--Amp\`{e}re equation

1. Introduction.

• The Monge–Ampère equation is fully nonlinear because its Hessian determinant forms the nonlinearity, and classical solutions may fail to exist even for smooth data.Alternative solution notions include Aleksandrov and viscosity solutions.Source: §1 · p. 1
• The model is a singularly perturbed fourth-order semilinear equation used as a regularization framework for computing viscosity solutions of the Monge–Ampère equation.In the radially symmetric setting, its solution converges uniformly in the viscosity sense to the Monge–Ampère solution as ϵ tends to zero.Source: §1 · p. 2
• Linear elements for the model equation pose stability and consistency challenges; prior method requires assumptions to ensure stability.Many earlier approaches to the biharmonic operator use at least quadratic elements.Source: §1 · p. 2
• Compared with the recovery-based finite element method in, this method achieves O(h2) convergence in L2 and O(h) in H1, versus O(h) and O(h0.5), respectively.Source: §1 · p. 2
• The discrete Miranda–Talenti estimate guides scheme design and stability analysis, while a fixed-point argument establishes existence, uniqueness, and a discrete H2 error estimate.A duality argument provides the optimal H1 error estimate.Source: §1 · p. 2
• The method uses a gradient recovery operator built through a Morley finite element function, whose true-gradient construction supports consistency.The operator and discrete Miranda–Talenti estimate build on prior work.Source: §1 · p. 2

2.1. Notation.

• The notation subsection specifies Sobolev-space conventions, mesh and edge entities, and the C0 linear, Crouzeix–Raviart, and Morley finite element spaces.It also defines nodal interpolation and lists trace, inverse, and Bramble–Hilbert inequalities.Source: §2.1 · pp. 3–4

2.3. Some technical results.

• The technical results collect inequalities and estimates, including trace and inverse bounds, that are used in subsequent analysis.The section also introduces a projection from L1(e) to P0(e).Source: §2.1 · p. 4, §2.3 · pp. 5, 7

2.2. Recovery operator.

• For each vh in Vh, the construction associates wh in Mh and defines Ghvh elementwise as ∇wh, the piecewise gradient of that associated function.The stated approximation-property assumption depends only on the mesh and holds on most commonly used meshes [14, Section 5.1].Source: §2.2 · pp. 4–5

3. Linearization and finite element approximation.

• Convexity of uϵ makes the cofactor matrix positive semidefinite, and the weak linearized problem is well-posed by the Lax–Milgram lemma.The weak formulation includes a boundary term involving the normal derivative.Source: §3 · p. 7
• The discrete scheme is designed from the discrete Miranda–Talenti estimate and uses the recovery operator in its bilinear form and boundary term.The formulation is given in (3.4).Source: §3 · p. 8
• Lax–Milgram establishes well-posedness of the discrete linearized scheme, and Theorem 3.3 gives an error estimate for solutions in H3(Ω) satisfying (3.1).The error analysis decomposes consistency contributions using recovered gradients and interpolation.Source: §3 · pp. 8–9

4.1. Numerical scheme.

• The scheme approximates the Hessian by derivatives of the recovered gradient, and its nonlinear term motivates a separate well-posedness analysis.The formulation begins from the model equation’s weak form and incorporates nonhomogeneous boundary data.Source: §4.1 · pp. 9–10

4.2. Existence, uniqueness and H2 error estimate of the numerical scheme.

• For sufficiently small h, the scheme has a unique solution and satisfies an H2 error estimate, established through a fixed-point argument.The operator T is well defined, and its fixed point solves the discrete scheme.Source: §4.2 · pp. 10, 12

4.3. H1 error estimate.

• A duality argument yields an H1 error estimate, with its analysis depending on regularity of the auxiliary problem.The authors note that the estimate in (4.7) is not sharp for γ > 1/2 when Δuϵ ∈ H2(Ω).Source: §4.3 · pp. 13, 17

5.1. Numerical Test 1.

• With fixed small mesh size h = 0.01 and varying ϵ, the error u − uϵ_h decays as O(ϵ), O(ϵ^3/4), and O(ϵ^1/4) in L2, H1, and discrete H2 norms, respectively.Source: §5.1 · p. 17
• For h = √ϵ, fitted error curves closely match numerical errors, supporting this choice as optimal across the tested norms.The fitted exponents are 0.25 for discrete H2, 0.75 for H1, and 1 for L2.Source: §5.1 · p. 18, Figure 5.1 · p. 19

5.3. Numerical Test 3.

• For data yielding a nonsmooth viscosity solution, the computed solution captures its convexity.The test uses f = 1, g = 0, and boundary condition ψϵ ≡ ϵ.Source: §5.3 · p. 19, Figure 5.2 · p. 20

5.2. Numerical Test 2.

• On uniform meshes, discrete H2, H1, and L2 errors converge at O(h^0.5), O(h), and O(h^2), respectively; compared with the existing recovery-based finite element method for the Monge–Ampère equation [12], L2 and H1 rates improve.Source: §5.2 · p. 18
• On unstructured meshes, H1 and discrete H2 errors show convergence rates similar to those on uniform meshes.The passage does not report an unstructured-mesh L2 rate.Source: §5.2 · p. 19, Table 5.4 · p. 20

5.4. Numerical Test 4.

• For the low-regularity solution u(x, y) = |x| in H1, the reported errors decrease, but no convergence rate is expected.The experiment uses Ω = (−1, 1)^2 and ϵ = 0.001.Source: §5.4 · p. 20

• The paper establishes convergence in both the discrete H2 and H1 norms.Source: §6 · p. 20
• Table 5.5 presents numerical results for the Monge–Ampère equation with solution u = |x|.Source: Table 5.5 · p. 21
• The proposed method outperforms an existing recovery-based finite element method for the Monge–Ampère equation in accuracy and convergence rates.Source: §6 · p. 21

Appendix A. Properties of the recovery operator.

• The appendix lists properties of Gh and says their proofs can be found in.Source: Appendix A · p. 21
• Theorem A.3 states a local approximation property for v in W 2,p(Ω), 1 ≤ p ≤ ∞, followed by a global approximation property.Source: Appendix A · p. 21
• Theorem A.4 gives a global superconvergence result for Gh when the mesh satisfies the (ν, µ) condition, v is in H3(Ω), h ≤ e−2, and β ∈ [0, 2].Source: Appendix A · p. 21