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Meshfree method for solving the elliptic Monge-Ampe`\grave{\textrm{e}}re
equation with Dirichlet boundary**footnotemark: *

Zhiyong Liu Qiuyan Xu∗ [email protected] School of Mathematics and Statistics,
Ningxia University, Yinchuan, China, 750021
Abstract

We prove the convergence of meshfree method for solving the elliptic Monge-Ampe`\grave{\textrm{e}}re equation with Dirichlet boundary on the bounded domain. L2L^{2} error is obtained based on the kernel-based trial spaces generated by the compactly supported radial basis functions. We obtain the convergence result when the testing discretization is finer than the trial discretization. The convergence rate depend on the regularity of the solution, the smoothness of the computing domain, and the approximation of scaled kernel-based spaces. The presented convergence theory covers a wide range of kernel-based trial spaces including stationary approximation and non-stationary approximation. An extension to non-Dirichlet boundary condition is in a forthcoming paper.

keywords:
Monge-Ampe`\grave{{e}}re equation; Meshfree method; Radial basis functions; Convergence rate
t1t1footnotetext: The research is supported by the Natural Science Foundations of China (No.12061057,12202219), and the Natural Science Foundations of Ningxia Province (No.2023AAC05001,2023AAC03003).

1 Introduction

We consider the meshfree method for numerically solving the Monge-Ampe`\grave{\textrm{e}}re equation with Dirichlet boundary condition:

det(D2​u​(x))\displaystyle\det(D^{2}u(\textbf{x})) =\displaystyle= f​(x),in​Ω⊂ℝd,\displaystyle f(\textbf{x}),\quad\textrm{in}\ \Omega\subset\mathbb{R}^{d}, (1.1)
u​(x)\displaystyle u(\textbf{x}) =\displaystyle= g​(x),on​∂Ω,\displaystyle g(\textbf{x}),\quad\textrm{on}\ \partial\Omega, (1.2)

here Ω\Omega is a strictly convex polygonal domain with a smooth boundary ∂Ω\partial\Omega, ff is a strictly positive function, and det(D2​u)\det(D^{2}u) is the determinant of the Hessian matrix D2​uD^{2}u. This equation is usually augmented by the convexity constraint of uu, which implies that the equation is elliptic. The numerical solution of Monge-Ampe`\grave{\textrm{e}}re equation is an very active topic of research. Many researchers developed kinds of numerical methods and analyzed their convergence, such as the finite difference methods [1, 2], finite element methods [3, 4], and spline methods [5]. In order to design a certain type of numerical method which can avoid the tedious mesh generation and the domain integration and can be capable of dealing with any irregular distribution of nodes, some meshfree methods based radial basis functions have been researched, which is started from 2013 in the paper [6]. And from then on, a series of meshfree methods have been proposed for solving the elliptic Monge-Ampe`\grave{\textrm{e}}re equation [7, 8, 9, 10]. Unfortunately, the convergence theory of meshfree method has not been proven yet. The present paper will prove the convergence of meshfree method for the Monge-Ampe`\grave{\textrm{e}}re equation. To simplify the proof, we only consider the case of taking d=2d=2:

F​u:=det(D2​u​(x,y))\displaystyle Fu:=\det(D^{2}u(x,y)) =\displaystyle= f​(x,y),in​Ω⊂ℝ2,\displaystyle f(x,y),\quad\textrm{in}\ \Omega\subset\mathbb{R}^{2}, (1.3)
u​(x,y)\displaystyle u(x,y) =\displaystyle= g​(x,y),on​∂Ω.\displaystyle g(x,y),\quad\textrm{on}\ \partial\Omega. (1.4)

We will rely the Bo¨\ddot{\textrm{o}}hmer/Schaback’s nonlinear discretization theory [11, 12] and the ideas of the latest paper [13], which discussed the convergence of unsymmetric radial basis functions method for second order quasilinear elliptic equations.

The paper is organized as follows. Section 2 is devoted to some notation and an introduction of kernel-based trial spaces generated by compactly supported radial basis functions. Section 3 introduces the well-posed theory of the Monge-Ampe`\grave{\textrm{e}}re equation and its meshfree discretization. Section 4 is the theoretical analysis of convergence.

2 Radial basis functions trial spaces

A translation-invariant kernel such as the strictly positive definite radial function Φ​(x,y)=Φ​(x−y)\Phi(\textbf{x},\textbf{y})=\Phi(\textbf{x}-\textbf{y}) will generate a reproducing kernel Hilbert space, the so-called native space 𝒩Φ​(Ω)\mathcal{N}_{\Phi}(\Omega). But we will make use of the native space 𝒩Φ​(ℝd)\mathcal{N}_{\Phi}(\mathbb{R}^{d}) which defined in ℝd\mathbb{R}^{d} and described by Fourier transform. 𝒩Φ​(ℝd)\mathcal{N}_{\Phi}(\mathbb{R}^{d}) consists of all functions f∈L2​(ℝd)f\in L^{2}(\mathbb{R}^{d}) with

‖f‖Φ2=∫ℝd|f^​(𝝎)|2Φ^​(𝝎)​𝑑𝝎<∞.\|f\|_{\Phi}^{2}=\int_{\mathbb{R}^{d}}\frac{|\widehat{f}(\boldsymbol{\omega})|^{2}}{\widehat{\Phi}(\boldsymbol{\omega})}d\boldsymbol{\omega}<\infty. (2.1)

Hence, if the Fourier transform of Φ:ℝd→ℝ\Phi:\mathbb{R}^{d}\rightarrow\mathbb{R} has the algebraic decay condition

c1​(1+‖𝝎‖22)−σ≤Φ^​(𝝎)≤c2​(1+‖𝝎‖22)−σc_{1}(1+\|\boldsymbol{\omega}\|_{2}^{2})^{-\sigma}\leq\widehat{\Phi}(\boldsymbol{\omega})\leq c_{2}(1+\|\boldsymbol{\omega}\|_{2}^{2})^{-\sigma} (2.2)

for two fixed constants 0<c1<c20<c_{1}<c_{2}, then the native space 𝒩Φ​(ℝd)\mathcal{N}_{\Phi}(\mathbb{R}^{d}) will be norm equivalent to the Sobolev space Wσ,2​(ℝd)W^{\sigma,2}(\mathbb{R}^{d}). Thus, with a radial function Φ\Phi satisfied (2.2) and a scaling parameter δ∈(0,1]\delta\in(0,1] we can construct a scaled radial function

Φδ:=δ−d​Φ​((x−y)/δ),x,y∈ℝd.\Phi_{\delta}:=\delta^{-d}\Phi((\textbf{x}-\textbf{y})/\delta),\quad\textbf{x},\textbf{y}\in\mathbb{R}^{d}. (2.3)

Thus we can consider the finite dimensional kernel-based trial space generated by Φδ\Phi_{\delta}. Let YI={y1,y2,…,yNI}⊆ΩY^{I}=\{\textbf{y}_{1},\textbf{y}_{2},\ldots,\textbf{y}_{N_{I}}\}\subseteq\Omega and YB={yNI+1,yNI+2,…,yN}⊆∂ΩY^{B}=\{\textbf{y}_{N_{I}+1},\textbf{y}_{N_{I}+2},\ldots,\textbf{y}_{N}\}\subseteq\partial\Omega be some data sites which are arranged on the domain and its boundary respectively. Then the whole of centers are Y=YI∪YBY=Y^{I}\cup Y^{B}, which can used to construct a trial space and determine its degrees of freedom. We denote B=B​(0,1)B=B(\textbf{0},1) as the unit ball in ℝd−1\mathbb{R}^{d-1} defined by using a number of Ck,γ−C^{k,\gamma}-diffeomorphisms ψj:B→Vj,1≤j≤K,\psi_{j}:B\rightarrow V_{j},\quad 1\leq j\leq K, and Vj⊆ℝdV_{j}\subseteq\mathbb{R}^{d} are open sets such that ∂Ω=⋃j=1KVj\partial\Omega=\bigcup_{j=1}^{K}V_{j}. Some measures are defined as follows:

hI:=supy∈Ωminyj∈YI⁡‖y−yj‖2,qI:=12​minyi,yk∈YI,i≠k⁡‖yi−yk‖2,for interior centers​YI,h_{I}:=\sup_{\textbf{y}\in\Omega}\min_{\textbf{y}_{j}\in Y^{I}}\|\textbf{y}-\textbf{y}_{j}\|_{2},\ q_{I}:=\frac{1}{2}\min_{\textbf{y}_{i},\textbf{y}_{k}\in Y^{I},i\neq k}\|\textbf{y}_{i}-\textbf{y}_{k}\|_{2},\quad\textrm{for interior centers}\ Y^{I},
hB:=max1≤j≤K⁡hB​j=max1≤j≤K⁡{supy∈B=B​(0,1)minyj∈ψj−1​(YB​⋂Vj)⁡‖y−yj‖2},for boundary centers​YB,h_{B}:=\max_{1\leq j\leq K}h_{Bj}=\max_{1\leq j\leq K}\left\{\sup_{\textbf{y}\in B=B(\textbf{0},1)}\min_{\textbf{y}_{j}\in\psi_{j}^{-1}\left(Y^{B}\bigcap V_{j}\right)}\|\textbf{y}-\textbf{y}_{j}\|_{2}\right\},\quad\textrm{for boundary centers}\ Y^{B},
qB:=min1≤j≤K⁡qB​j=min1≤j≤K⁡{12​minyi,yk∈ψj−1​(YB​⋂Vj),i≠k⁡‖yi−yk‖2},for boundary centers​YB,q_{B}:=\min_{1\leq j\leq K}q_{Bj}=\min_{1\leq j\leq K}\left\{\frac{1}{2}\min_{\textbf{y}_{i},\textbf{y}_{k}\in\psi_{j}^{-1}\left(Y^{B}\bigcap V_{j}\right),i\neq k}\|\textbf{y}_{i}-\textbf{y}_{k}\|_{2}\right\},\quad\textrm{for boundary centers}\ Y^{B},
hY:=max⁡{hI,hB}<1,qY:=min⁡{qI,qB}.h_{Y}:=\max\{h_{I},h_{B}\}<1,\quad q_{Y}:=\min\{q_{I},q_{B}\}.

Using the scaled radial function Φδ\Phi_{\delta} as basis functions, then the radial basis functions trial space is

Wδ=span{Φδ(⋅−y)|y∈Y}.W_{\delta}=\textrm{span}\{\Phi_{\delta}(\cdot-\textbf{y})|\textbf{y}\in Y\}. (2.4)

In the later theoretical analysis, we need two important auxiliary results: inverse inequality and sampling inequality. The following inverse inequality has been proved for RBF approximation in the reference [13] on the bounded domains.

Lemma 2.1.

(inverse inequality) Let Ω⊆ℝd\Omega\subseteq\mathbb{R}^{d} be a bounded domain satisfying an interior cone condtion. Let YY be quasi-uniform in the sense that hY≤c​qYh_{Y}\leq cq_{Y}. For all δ−\delta-dependent functions of the form f=∑j=1NcjΦδ(⋅−xj),xj∈Y,f=\sum_{j=1}^{N}c_{j}\Phi_{\delta}(\cdot-\textbf{x}_{j}),\ \textbf{x}_{j}\in Y, there exists a constant CC such that

‖f‖Wσ,2​(Ω)≤C​δσ​hY−σ+d/2​‖f‖L2​(Ω).\|f\|_{W^{\sigma,2}(\Omega)}\leq C\delta^{\sigma}h_{Y}^{-\sigma+d/2}\|f\|_{L^{2}(\Omega)}. (2.5)

The well-known sampling inequality has been proved in [14, 15].

Lemma 2.2.

(sampling inequality) Suppose Ω⊆ℝd\Omega\subseteq\mathbb{R}^{d} is a bounded domain with an interior cone condition. Let q∈[1,∞]q\in[1,\infty] and constants 0≤μ<μ+d/2<⌊m⌋0\leq\mu<\mu+d/2<\lfloor m\rfloor. Then there are positive constant CC, h0h_{0} such that

‖f‖Wμ,q​(Ω)≤C​(hYm−μ−d​(1/2−1/q)+​‖f‖Wm,2​(Ω)+hY−μ​‖ΠY​f‖∞,Y)\|f\|_{W^{\mu,q}(\Omega)}\leq C\left(h_{Y}^{m-\mu-d(1/2-1/q)_{+}}\|f\|_{W^{m,2}(\Omega)}+h_{Y}^{-\mu}\|\Pi_{Y}f\|_{\infty,Y}\right) (2.6)

holds for every discrete set Y⊂ΩY\subset\Omega with fill distance at most hY≤h0h_{Y}\leq h_{0} and every f∈Wm,2​(Ω)f\in W^{m,2}(\Omega). Here ΠY\Pi_{Y} is the function evaluation operator at the data sites YY.

3 Meshfree discretization

3.1 Well-posedness

There are several theories which describe the regularity and the well-posedness of the Monge-Ampe`\grave{\textrm{e}}re equation [16, 17, 18], for example the existence theorem from [18]:

Theorem 3.1.

(existence) Let Ω\Omega be a uniformly convex domain with Ck+2,γ−C^{k+2,\gamma}-boundary, k≥2,γ∈(0,1)k\geq 2,\gamma\in(0,1). Let f∈Ck,γ​(Ω¯)f\in C^{k,\gamma}(\bar{\Omega}) be strictly positive. Then for any g∈Ck+2,γ​(∂Ω)g\in C^{k+2,\gamma}(\partial\Omega), there exist a unique solution u∗∈Ck+2,γ​(Ω¯)u^{*}\in C^{k+2,\gamma}(\bar{\Omega}) to the Dirichlet problem (1.1)-(1.2).

In the later discussion, we always assume that the conditions of Theorem 3.1 are satisfied. And we assume that the solution of the equations (1.3)-(1.4) has the higher regularity u∗∈Wσ,2​(Ω)u^{*}\in W^{\sigma,2}(\Omega) with σ\sigma such that ⌊σ⌋>k+2+d/2\lfloor\sigma\rfloor>k+2+d/2. Of course, u∗∈Ck+2,γ​(Ω¯)u^{*}\in C^{k+2,\gamma}(\overline{\Omega}) according to Sobolev inequality from section 5.6.3 of [19]. In addition, FF is Fre´\acute{\textrm{e}}chet differentiable in KR​(u∗)={u∈Wσ,2​(Ω):‖u−u∗‖Wσ,2​(Ω)≤R}⊂D​(F)K_{R}(u^{*})=\{u\in W^{\sigma,2}(\Omega):\|u-u^{*}\|_{W^{\sigma,2}(\Omega)}\leq R\}\subset D(F), and

F′​(u∗)​(v)=uy​y∗​vx​x+ux​x∗​vy​y−2​ux​y∗​vx​y.F^{\prime}(u^{*})(v)=u^{*}_{yy}v_{xx}+u^{*}_{xx}v_{yy}-2u^{*}_{xy}v_{xy}. (3.1)

Obviously, the convex property of u∗u^{*} implies that the linear operator F′​(u∗)F^{\prime}(u^{*}) is uniformly elliptic. Thus, we can obtain a prior bounds of the fully nonlinear problem (1.3)-(1.4) by utilizing the prior estimate of the linear elliptic problem

F′​(u)​v​(x,y)\displaystyle F^{\prime}(u)v(x,y) =\displaystyle= f​(x,y),in​Ω⊂ℝ2,\displaystyle f(x,y),\quad\textrm{in}\ \Omega\subset\mathbb{R}^{2}, (3.2)
v​(x,y)\displaystyle v(x,y) =\displaystyle= g​(x,y),on​∂Ω,\displaystyle g(x,y),\quad\textrm{on}\ \partial\Omega, (3.3)

here u∈KR​(u∗)u\in K_{R}(u^{*}).

Theorem 3.2.

(a priori bounds of (1.3)-(1.4)) Suppose Ω⊆ℝd,d=2\Omega\subseteq\mathbb{R}^{d},d=2 is a uniformly convex domain with Ck+2,γ−C^{k+2,\gamma}-boundary. Then the Freche´\acute{\textrm{e}}t derivative F′​(u)F^{\prime}(u) at uu be bounded and Lipschitz continuous, namely

‖F′​(u)−F′​(v)‖:=‖F′​(u∗)​u−F′​(u∗)​v‖L2​(Ω)≤C′′​‖u−v‖Wσ,2​(Ω),for all​u,v∈KR​(u∗).\|F^{\prime}(u)-F^{\prime}(v)\|:=\|F^{\prime}(u^{*})u-F^{\prime}(u^{*})v\|_{L^{2}(\Omega)}\leq C^{\prime\prime}\|u-v\|_{W^{\sigma,2}(\Omega)},\quad\textrm{for all}\ u,v\in K_{R}(u^{*}). (3.4)

Assume the radius RR to be small enough to satisfy

Cl​C′′​R≤12,C_{l}C^{\prime\prime}R\leq\frac{1}{2}, (3.5)

then for any u,v∈KR​(u∗)u,v\in K_{R}(u^{*}), there exists two constants C,CbC,C_{b} such that

‖u−v‖L2​(Ω)≤C​{‖F​u−F​v‖L2​(Ω)+‖u−v‖W1/2,2​(∂Ω)},\|u-v\|_{L^{2}(\Omega)}\leq C\{\|Fu-Fv\|_{L^{2}(\Omega)}+\|u-v\|_{W^{1/2,2}(\partial\Omega)}\}, (3.6)
‖F​u−F​v‖Wσ−2,2​(Ω)≤Cb​‖u−v‖Wσ,2​(Ω).\|Fu-Fv\|_{W^{\sigma-2,2}(\Omega)}\leq C_{b}\|u-v\|_{W^{\sigma,2}(\Omega)}. (3.7)
Proof.

The inequality (3.4) is obvious because ‖F′​(u)−F′​(v)‖\|F^{\prime}(u)-F^{\prime}(v)\| can be bounded by

‖F′​(u∗)​u−F′​(u∗)​v‖L2​(Ω)\displaystyle\|F^{\prime}(u^{*})u-F^{\prime}(u^{*})v\|_{L^{2}(\Omega)} =‖ux​x∗​(u−v)y​y+uy​y∗​(u−v)x​x−2​ux​y∗​(u−v)x​y‖L2​(Ω)\displaystyle=\|u^{*}_{xx}(u-v)_{yy}+u^{*}_{yy}(u-v)_{xx}-2u^{*}_{xy}(u-v)_{xy}\|_{L^{2}(\Omega)}
≤|u∗|W2,2​(Ω)​|u−v|W2,2​(Ω)\displaystyle\leq|u^{*}|_{W^{2,2}(\Omega)}|u-v|_{W^{2,2}(\Omega)}
≤C′′​‖u−v‖Wσ,2​(Ω).\displaystyle\leq C^{{}^{\prime\prime}}\|u-v\|_{W^{\sigma,2}(\Omega)}.

With the help of the priori inequality of linear elliptic problems (for example Corollary 8.7 of [20],), we have

‖u−v‖L2​(Ω)≤Cl​{‖F′​(u)​(u−v)‖L2​(Ω)+‖u−v‖W1/2,2​(∂Ω)}.\|u-v\|_{L^{2}(\Omega)}\leq C_{l}\{\|F^{\prime}(u)(u-v)\|_{L^{2}(\Omega)}+\|u-v\|_{W^{1/2,2}(\partial\Omega)}\}.

Then by using Taylor formula and Lipschitz continuity of F′​(u)F^{\prime}(u), we have

‖u−v‖L2​(Ω)\displaystyle\|u-v\|_{L^{2}(\Omega)} ≤Cl​{‖F′​(u)​(u−v)−F​u+F​v‖L2​(Ω)+‖F​u−F​v‖L2​(Ω)+‖u−v‖W1/2,2​(∂Ω)}\displaystyle\leq C_{l}\{\|F^{\prime}(u)(u-v)-Fu+Fv\|_{L^{2}(\Omega)}+\|Fu-Fv\|_{L^{2}(\Omega)}+\|u-v\|_{W^{1/2,2}(\partial\Omega)}\}
≤Cl​{supw∈u​v¯‖F′​(w)−F′​(v)‖​‖u−v‖L2​(Ω)+‖F​u−F​v‖L2​(Ω)+‖u−v‖W1/2,2​(∂Ω)}\displaystyle\leq C_{l}\{\sup_{w\in\overline{uv}}\|F^{\prime}(w)-F^{\prime}(v)\|\|u-v\|_{L^{2}(\Omega)}+\|Fu-Fv\|_{L^{2}(\Omega)}+\|u-v\|_{W^{1/2,2}(\partial\Omega)}\}
≤Cl​{C′′‖u−v∥Wσ,2​(Ω)​‖u−v‖L2​(Ω)+‖F​u−F​v‖L2​(Ω)+‖u−v‖W1/2,2​(∂Ω)}\displaystyle\leq C_{l}\{C^{\prime\prime}\|u-v\|_{W^{\sigma,2}(\Omega)}\|u-v\|_{L^{2}(\Omega)}+\|Fu-Fv\|_{L^{2}(\Omega)}+\|u-v\|_{W^{1/2,2}(\partial\Omega)}\}
≤Cl​{C′′​R‖u−v∥L2​(Ω)+‖F​u−F​v‖L2​(Ω)+‖u−v‖W1/2,2​(∂Ω)}.\displaystyle\leq C_{l}\{C^{\prime\prime}R\|u-v\|_{L^{2}(\Omega)}+\|Fu-Fv\|_{L^{2}(\Omega)}+\|u-v\|_{W^{1/2,2}(\partial\Omega)}\}.

This yields the result (3.6). In addition,

‖F​u−F​v‖Wσ−2,2​(Ω)\displaystyle\|Fu-Fv\|_{W^{\sigma-2,2}(\Omega)} ≤‖F​v−F​u−F′​(u)​(v−u)‖Wσ−2,2​(Ω)+‖F′​(u)​(v−u)‖Wσ−2,2​(Ω)\displaystyle\leq\|Fv-Fu-F^{\prime}(u)(v-u)\|_{W^{\sigma-2,2}(\Omega)}+\|F^{\prime}(u)(v-u)\|_{W^{\sigma-2,2}(\Omega)}
≤C′′​R​‖u−v‖Wσ,2​(Ω)+(C′′​R+‖F′​(u∗)‖)​‖u−v‖Wσ,2​(Ω)\displaystyle\leq C^{\prime\prime}R\|u-v\|_{W^{\sigma,2}(\Omega)}+(C^{\prime\prime}R+\|F^{\prime}(u^{*})\|)\|u-v\|_{W^{\sigma,2}(\Omega)}
≤(2​C′′​R+‖F′​(u∗)‖)​‖u−v‖Wσ,2​(Ω).\displaystyle\leq(2C^{\prime\prime}R+\|F^{\prime}(u^{*})\|)\|u-v\|_{W^{\sigma,2}(\Omega)}.

Setting Cb=2​C′′​R+‖F′​(u∗)‖C_{b}=2C^{\prime\prime}R+\|F^{\prime}(u^{*})\| yields (3.7). ∎

3.2 Collocation

We are now consider the meshfree collocation for the Monge-Ampe`\grave{\textrm{e}}re equation (1.3)-(1.4). Denote x=(x,y)\textbf{x}=(x,y) and xi=(xi,yi)\textbf{x}_{i}=(x_{i},y_{i}), we use a set X=XI∪XBX=X^{I}\cup X^{B} with XI={x1,x2,…,xMI}⊆ΩX^{I}=\{\textbf{x}_{1},\textbf{x}_{2},\ldots,\textbf{x}_{M_{I}}\}\subseteq\Omega and XB={xMI+1,xMI+2,…,xM}⊆∂ΩX^{B}=\{\textbf{x}_{M_{I}+1},\textbf{x}_{M_{I}+2},\ldots,\textbf{x}_{M}\}\subseteq\partial\Omega as collocation points, and use

sI:=supx∈Ωminxj∈XI⁡‖x−xj‖2,s_{I}:=\sup_{\textbf{x}\in\Omega}\min_{\textbf{x}_{j}\in X^{I}}\|\textbf{x}-\textbf{x}_{j}\|_{2},
sB:=max1≤j≤K⁡sB​j=max1≤j≤K⁡{supx∈B=B​(0,1)minxj∈ψj−1​(XB​⋂Vj)⁡‖x−xj‖2}s_{B}:=\max_{1\leq j\leq K}s_{Bj}=\max_{1\leq j\leq K}\left\{\sup_{\textbf{x}\in B=B(\textbf{0},1)}\min_{\textbf{x}_{j}\in\psi_{j}^{-1}\left(X^{B}\bigcap V_{j}\right)}\|\textbf{x}-\textbf{x}_{j}\|_{2}\right\}

to represent the fill distance of the data sites XIX^{I} and XBX^{B} respectively. Obviously, they have the parallel definitions with hIh_{I} and hBh_{B} on trial side. Hence choose sX=max⁡{sI,sB}<1s_{X}=\max\{s_{I},s_{B}\}<1. Then we can construct a approximate function with the form

s​(x)=∑j=1Ncj​Φδ​(x−yj),yj∈Y=YI∪YB⊂ℝ2,s(\textbf{x})=\sum_{j=1}^{N}c_{j}\Phi_{\delta}(\textbf{x}-\textbf{y}_{j}),\quad\textbf{y}_{j}\in Y=Y^{I}\cup Y^{B}\subset\mathbb{R}^{2},

which satisfies

F​s​(xi,yi)\displaystyle Fs(x_{i},y_{i}) =f​(xi,yi),(xi,yi)∈XI,i=1,…,MI,\displaystyle=f(x_{i},y_{i}),\quad(x_{i},y_{i})\in X^{I},\quad i=1,\ldots,M_{I}, (3.8)
s​(xi,yi)\displaystyle s(x_{i},y_{i}) =g​(xi,yi),(xi,yi)∈XB,i=MI+1,…,M.\displaystyle=g(x_{i},y_{i}),\quad(x_{i},y_{i})\in X^{B},\quad i=M_{I+1},\ldots,M. (3.9)

(3.8)-(3.9) is a meshfree discretization of the equation (1.3)-(1.4), but with a nonsquare nonlinear systems. Let ΠXI\Pi_{X^{I}} and ΠXB\Pi_{X^{B}} be the function evaluation operators at the data sites XIX^{I} and XBX^{B} respectively, then the nonlinear systems (3.8)-(3.9) can also be rewritten as

ΠXI​F​s\displaystyle\Pi_{X^{I}}Fs =ΠXI​f=ΠXI​F​u∗,\displaystyle=\Pi_{X^{I}}f=\Pi_{X^{I}}Fu^{*}, (3.10)
ΠXB​s\displaystyle\Pi_{X^{B}}s =ΠXB​g=ΠXB​u∗.\displaystyle=\Pi_{X^{B}}g=\Pi_{X^{B}}u^{*}. (3.11)

Because the nonlinear systems (3.10)-(3.11) is overdetermined, thus we only require that one of the nonlinear numerical methods is clever enough to produce a s∈Wδ∩KR​(u∗)s\in W_{\delta}\cap K_{R}(u^{*}) with the given tolerance

max⁡{‖ΠXI​(F​u∗−F​s)‖∞,XI,‖ΠXB​(u∗−s)‖∞,XB}≤tol.\max\left\{\|\Pi_{X^{I}}(Fu^{*}-Fs)\|_{\infty,X^{I}},\|\Pi_{X^{B}}(u^{*}-s)\|_{\infty,X^{B}}\right\}\leq\emph{tol}. (3.12)

This can be done by residual minimization

s=arg⁡min⁡{‖ΠXI​(F​u∗−F​s)‖∞,XI,‖ΠXB​(u∗−s)‖∞,XB:s∈Wδ∩KR​(u∗)}.s=\arg\min\left\{\|\Pi_{X^{I}}(Fu^{*}-Fs)\|_{\infty,X^{I}},\|\Pi_{X^{B}}(u^{*}-s)\|_{\infty,X^{B}}:s\in W_{\delta}\cap K_{R}(u^{*})\right\}. (3.13)

4 Convergence analysis

The following theorem is our main theoretical result which proves the convergence rate of meshfree method for the Monge-Ampe`\grave{\textrm{e}}re equation (1.3)-(1.4).

Theorem 4.1.

Let Ω⊆ℝd,d=2\Omega\subseteq\mathbb{R}^{d},d=2 be a uniformly convex domain has a Ck+2,γ−C^{k+2,\gamma}-boundary with γ∈[0,1)\gamma\in[0,1). In the equation (1.3)-(1.4), let f∈Ck,γ​(Ω¯)f\in C^{k,\gamma}(\bar{\Omega}) be strictly positive and g∈Ck+2,γ​(∂Ω)g\in C^{k+2,\gamma}(\partial\Omega). We denote the unique solution of (1.3)-(1.4) as u∗∈Wσ,2​(Ω)u^{*}\in W^{\sigma,2}(\Omega), ⌊σ⌋>k+2+d/2\lfloor\sigma\rfloor>k+2+d/2. Assume the radius RR of the ball KRK_{R} to be small enough to satisfy (3.5). Let YY be quasi-uniform in the sense that hY≤c​qYh_{Y}\leq cq_{Y}. Let the collocation sites XX also be quasi-uniform. Let sIs_{I} and sBs_{B} be the fill distance of the data sites XIX^{I} and XBX^{B} respectively, and sX=max⁡{sI,sB}s_{X}=\max\{s_{I},s_{B}\}. Suppose s∈Wδs\in W_{\delta} is the collocation solution of (3.8)-(3.9) obtained by choosing

t​o​l=C⋅sB1/2​δ−σ​hYσ−2−d/2​‖u∗‖Wσ,2​(Ω).tol=C\cdot s^{1/2}_{B}\delta^{-\sigma}h_{Y}^{\sigma-2-d/2}\|u^{*}\|_{W^{\sigma,2}(\Omega)}. (4.1)

If the testing discretizations is finer than the trial discretization and satisfies condition

C⋅Cb⋅δσ​sXσ−2​hY−σ+d/2<12,C\cdot C_{b}\cdot\delta^{\sigma}s_{X}^{\sigma-2}h_{Y}^{-\sigma+d/2}<\frac{1}{2}, (4.2)

then there exists a constant C>0C>0 such that

‖u∗−s‖L2​(Ω)≤C​δ−σ​hYσ−3​‖u∗‖Wσ,2​(Ω).\|u^{*}-s\|_{L^{2}(\Omega)}\leq C\delta^{-\sigma}h^{\sigma-3}_{Y}\|u^{*}\|_{W^{\sigma,2}(\Omega)}.
Proof.

We want to bound the L2−L^{2}-error between the solution u∗u^{*} and its approximation ss. We can use the radial basis functions interpolant as a transition. The interpolant is a map Ih:u∗→Ih​u∗∈WδI_{h}:u^{*}\rightarrow I_{h}u^{*}\in W_{\delta} defined by

Ih​u∗​(yk)=∑j=1Ncj​Φδ​(yk−yj)=u∗​(yk),yk∈Y⊂ℝ2,k=1,2,…,N,I_{h}u^{*}(\textbf{y}_{k})=\sum_{j=1}^{N}c_{j}\Phi_{\delta}(\textbf{y}_{k}-\textbf{y}_{j})=u^{*}(\textbf{y}_{k}),\quad\textbf{y}_{k}\in Y\subset\mathbb{R}^{2},k=1,2,\ldots,N,

for the unknown coefficients c1,c2,…,cNc_{1},c_{2},\ldots,c_{N} are determined by solving the above linear system. Then the L2−L^{2}-error between the solution u∗u^{*} and its approximation ss can be split into

‖u∗−s‖L2​(Ω)\displaystyle\|u^{*}-s\|_{L^{2}(\Omega)} ≤‖u∗−Ih​u∗‖L2​(Ω)+‖Ih​u∗−s‖L2​(Ω).\displaystyle\leq\|u^{*}-I_{h}u^{*}\|_{L^{2}(\Omega)}+\|I_{h}u^{*}-s\|_{L^{2}(\Omega)}.

The first term on the right-hand side is the radial basis functions interpolation error, which can be easily obtained by using the sampling inequality (Lemma 2.2) and the stability of IhI_{h} (Lemma 5.1 of [13]), namely

‖u∗−Ih​u∗‖L2​(Ω)≤C​hYσ​‖u∗−Ih​u∗‖Wσ,2​(Ω)≤C​δ−σ​hYσ​‖u∗‖Wσ,2​(Ω).\displaystyle\|u^{*}-I_{h}u^{*}\|_{L^{2}(\Omega)}\leq Ch_{Y}^{\sigma}\|u^{*}-I_{h}u^{*}\|_{W^{\sigma,2}(\Omega)}\leq C\delta^{-\sigma}h_{Y}^{\sigma}\|u^{*}\|_{W^{\sigma,2}(\Omega)}. (4.3)

Hence, our main task in this proof is to bound the second term on the right-hand side.

Choosing u=Ih​u∗,v=su=I_{h}u^{*},v=s and squaring both sides of (3.6), we have

‖Ih​u∗−s‖L2​(Ω)2≤2​C​{‖F​(Ih​u∗)−F​s‖L2​(Ω)2+‖Ih​u∗−s‖W1/2,2​(∂Ω)2}.\|I_{h}u^{*}-s\|_{L^{2}(\Omega)}^{2}\leq 2C\{\|F(I_{h}u^{*})-Fs\|_{L^{2}(\Omega)}^{2}+\|I_{h}u^{*}-s\|_{W^{1/2,2}(\partial\Omega)}^{2}\}.

Then we can bound the two terms on the right-hand side separately. For the first term, using Lemma 2.2, the inequality (3.7), Lemma 2.1, and the stability of radial basis functions interpolation (Lemma 5.1 of [13]) yields

‖F​(Ih​u∗)−F​s‖L2​(Ω)2\displaystyle\|F(I_{h}u^{*})-Fs\|_{L^{2}(\Omega)}^{2} ≤C​{sI2​σ−4‖F​(Ih​u∗)−F​s∥Wσ−2,2​(Ω)2+‖ΠXI​(F​(Ih​u∗)−F​s)‖∞,XI2}\displaystyle\leq C\left\{s_{I}^{2\sigma-4}\|F(I_{h}u^{*})-Fs\|^{2}_{W^{\sigma-2,2}(\Omega)}+\|\Pi_{X^{I}}(F(I_{h}u^{*})-Fs)\|^{2}_{\infty,X^{I}}\right\}
≤C⋅Cb2⋅sI2​σ−4​‖Ih​u∗−s‖Wσ,2​(Ω)2+C​‖ΠXI​(F​(Ih​u∗)−F​u∗)‖∞,XI2+C⋅t​o​l2\displaystyle\leq C\cdot C_{b}^{2}\cdot s_{I}^{2\sigma-4}\|I_{h}u^{*}-s\|^{2}_{W^{\sigma,2}(\Omega)}+C\|\Pi_{X^{I}}(F(I_{h}u^{*})-Fu^{*})\|^{2}_{\infty,X^{I}}+C\cdot tol^{2}
=C⋅Cb2⋅sI2​σ−4​‖Ih​u∗−s‖Wσ,2​(Ω)2+C​‖F​(Ih​u∗)−F​u∗‖∞,XI2+C⋅t​o​l2\displaystyle=C\cdot C_{b}^{2}\cdot s_{I}^{2\sigma-4}\|I_{h}u^{*}-s\|^{2}_{W^{\sigma,2}(\Omega)}+C\|F(I_{h}u^{*})-Fu^{*}\|^{2}_{\infty,X^{I}}+C\cdot tol^{2}
≤C⋅Cb2⋅δ2​σ​sI2​σ−4​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​‖F​(Ih​u∗)−F​u∗‖∞,Ω2+C⋅t​o​l2\displaystyle\leq C\cdot C_{b}^{2}\cdot\delta^{2\sigma}s_{I}^{2\sigma-4}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\|F(I_{h}u^{*})-Fu^{*}\|^{2}_{\infty,\Omega}+C\cdot tol^{2}
≤C⋅Cb2⋅δ2​σ​sX2​σ−4​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​‖Ih​u∗−u∗‖W2,∞​(Ω)2+C⋅t​o​l2\displaystyle\leq C\cdot C_{b}^{2}\cdot\delta^{2\sigma}s_{X}^{2\sigma-4}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\|I_{h}u^{*}-u^{*}\|^{2}_{W^{2,\infty}(\Omega)}+C\cdot tol^{2}
≤C⋅Cb2⋅δ2​σ​sX2​σ−4​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​hI2​(σ−2−d/2)​‖Ih​u∗−u∗‖Wσ,2​(Ω)2+C⋅t​o​l2\displaystyle\leq C\cdot C_{b}^{2}\cdot\delta^{2\sigma}s_{X}^{2\sigma-4}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Ch_{I}^{2(\sigma-2-d/2)}\|I_{h}u^{*}-u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}+C\cdot tol^{2}
≤C⋅Cb2⋅δ2​σ​sX2​σ−4​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​δ−2​σ​hI2​(σ−2−d/2)​‖u∗‖Wσ,2​(Ω)2+C⋅t​o​l2\displaystyle\leq C\cdot C_{b}^{2}\cdot\delta^{2\sigma}s_{X}^{2\sigma-4}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\delta^{-2\sigma}h_{I}^{2(\sigma-2-d/2)}\|u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}+C\cdot tol^{2}
≤C⋅Cb2⋅δ2​σ​sX2​σ−4​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​δ−2​σ​hY2​(σ−2−d/2)​‖u∗‖Wσ,2​(Ω)2+C⋅t​o​l2\displaystyle\leq C\cdot C_{b}^{2}\cdot\delta^{2\sigma}s_{X}^{2\sigma-4}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\delta^{-2\sigma}h_{Y}^{2(\sigma-2-d/2)}\|u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}+C\cdot tol^{2}
≤C⋅Cb2⋅δ2​σ​sX2​σ−4​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​δ−2​σ​hY2​(σ−2−d/2)​‖u∗‖Wσ,2​(Ω)2.\displaystyle\leq C\cdot C_{b}^{2}\cdot\delta^{2\sigma}s_{X}^{2\sigma-4}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\delta^{-2\sigma}h_{Y}^{2(\sigma-2-d/2)}\|u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}.

For the boundary term, we denote wj=((Ih​u∗−s)​ωj)∘Ψj∈Wσ−1/2,2​(B)w_{j}=((I_{h}u^{*}-s)\omega_{j})\circ\Psi_{j}\in W^{\sigma-1/2,2}(B). From Lemma 2.2, the trace theorem (Theorem 8.7 of [21]), the Sobolev imbedding theorems (Theorem 4.12 of [22]), the quasi-uniform property of XX, and the quasi-uniform property of YY,we can bound

‖Ih​u∗−s‖W1/2,2​(∂Ω)2=∑j=1K‖wj‖W1/2,2​(B)2\displaystyle\quad\|I_{h}u^{*}-s\|^{2}_{W^{1/2,2}(\partial\Omega)}=\sum_{j=1}^{K}\|w_{j}\|^{2}_{W^{1/2,2}(B)}
≤C​∑j=1K(sB​jσ−1​‖wj‖Wσ−1/2,2​(B)+sB​j−1/2​‖Πψj−1​(XB​⋂Vj)​wj‖∞,ψj−1​(XB​⋂Vj))2\displaystyle\leq C\sum_{j=1}^{K}\left(s_{Bj}^{\sigma-1}\|w_{j}\|_{W^{\sigma-1/2,2}(B)}+s_{Bj}^{-1/2}\|\Pi_{\psi_{j}^{-1}\left(X^{B}\bigcap V_{j}\right)}w_{j}\|_{\infty,\psi_{j}^{-1}\left(X^{B}\bigcap V_{j}\right)}\right)^{2}
≤C​∑j=1K(sB​j2​σ−2​‖wj‖Wσ−1/2,2​(B)2+sB​j−1​‖Πψj−1​(XB​⋂Vj)​wj‖∞,ψj−1​(XB​⋂Vj)2)\displaystyle\leq C\sum_{j=1}^{K}\left(s_{Bj}^{2\sigma-2}\|w_{j}\|^{2}_{W^{\sigma-1/2,2}(B)}+s_{Bj}^{-1}\|\Pi_{\psi_{j}^{-1}\left(X^{B}\bigcap V_{j}\right)}w_{j}\|^{2}_{\infty,\psi_{j}^{-1}\left(X^{B}\bigcap V_{j}\right)}\right)
≤C​sB2​σ−2​‖Ih​u∗−s‖Wσ−1/2,2​(∂Ω)2+C​sB−1​‖ΠXB​(Ih​u∗−s)‖∞,XB2\displaystyle\leq Cs_{B}^{2\sigma-2}\|I_{h}u^{*}-s\|^{2}_{W^{\sigma-1/2,2}(\partial\Omega)}+Cs_{B}^{-1}\|\Pi_{X^{B}}(I_{h}u^{*}-s)\|^{2}_{\infty,X^{B}}
≤C​sB2​σ−2​‖Ih​u∗−s‖Wσ,2​(Ω)2+C​sB−1​‖ΠXB​(Ih​u∗−u∗)‖∞,XB2+C​sB−1⋅t​o​l2\displaystyle\leq Cs_{B}^{2\sigma-2}\|I_{h}u^{*}-s\|^{2}_{W^{\sigma,2}(\Omega)}+Cs_{B}^{-1}\|\Pi_{X^{B}}(I_{h}u^{*}-u^{*})\|^{2}_{\infty,X^{B}}+Cs_{B}^{-1}\cdot tol^{2}
≤C​δ2​σ​sX2​σ−2​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​sB−1​‖Ih​u∗−u∗‖∞,XB2+C​sB−1⋅t​o​l2\displaystyle\leq C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Cs_{B}^{-1}\|I_{h}u^{*}-u^{*}\|^{2}_{\infty,X^{B}}+Cs_{B}^{-1}\cdot tol^{2}
≤C​δ2​σ​sX2​σ−2​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​sB−1​max1≤j≤K⁡‖wj‖L∞​(B)2+C​sB−1⋅t​o​l2\displaystyle\leq C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Cs_{B}^{-1}\max_{1\leq j\leq K}\|w_{j}\|^{2}_{L^{\infty}(B)}+Cs_{B}^{-1}\cdot tol^{2}
≤C​δ2​σ​sX2​σ−2​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​sB−1​max1≤j≤K⁡hB​j2​(σ−1/2−d/2)​‖wj‖Wσ−1/2,2​(B)2+C​sB−1⋅t​o​l2\displaystyle\leq C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Cs_{B}^{-1}\max_{1\leq j\leq K}h_{Bj}^{2(\sigma-1/2-d/2)}\|w_{j}\|^{2}_{W^{\sigma-1/2,2}(B)}+Cs_{B}^{-1}\cdot tol^{2}
≤C​δ2​σ​sX2​σ−2​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​sB−1​hB2​(σ−1/2−d/2)​∑j=1K‖wj‖Wσ−1/2,2​(B)2+C​sB−1⋅t​o​l2\displaystyle\leq C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Cs_{B}^{-1}h_{B}^{2(\sigma-1/2-d/2)}\sum_{j=1}^{K}\|w_{j}\|^{2}_{W^{\sigma-1/2,2}(B)}+Cs_{B}^{-1}\cdot tol^{2}
=C​δ2​σ​sX2​σ−2​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​sB−1​hB2​(σ−1/2−d/2)​‖Ih​u∗−u∗‖Wσ−1/2,2​(∂Ω)2+C​sB−1⋅t​o​l2\displaystyle=C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Cs_{B}^{-1}h_{B}^{2(\sigma-1/2-d/2)}\|I_{h}u^{*}-u^{*}\|^{2}_{W^{\sigma-1/2,2}(\partial\Omega)}+Cs_{B}^{-1}\cdot tol^{2}
≤C​δ2​σ​sX2​σ−2​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​sB−1​hB2​(σ−1/2−d/2)​‖Ih​u∗−u∗‖Wσ,2​(Ω)2+C​sB−1⋅t​o​l2\displaystyle\leq C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Cs_{B}^{-1}h_{B}^{2(\sigma-1/2-d/2)}\|I_{h}u^{*}-u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}+Cs_{B}^{-1}\cdot tol^{2}
≤C​δ2​σ​sX2​σ−2​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​δ−2​σ​hY−(σ−d/2)/(σ−2)​hB2​(σ−1/2−d/2)​‖u∗‖Wσ,2​(Ω)2+C​sB−1⋅t​o​l2\displaystyle\leq C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\delta^{-2\sigma}h_{Y}^{-(\sigma-d/2)/(\sigma-2)}h_{B}^{2(\sigma-1/2-d/2)}\|u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}+Cs_{B}^{-1}\cdot tol^{2}
≤C​δ2​σ​sX2​σ−2​hY−2​σ+d​‖Ih​u∗−s‖L2​(Ω)2+C​δ−2​σ​hY2​(σ−2−d/2)​‖u∗‖Wσ,2​(Ω)2.\displaystyle\leq C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\delta^{-2\sigma}h_{Y}^{2(\sigma-2-d/2)}\|u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}.

By using the condition (4.2), combining both above results with the interpolation error (4.3) and setting d=2d=2, we obtain the convergence rate. ∎

5 Conclusions

Although the meshfree method has been used for solving the Monge-Ampe`\grave{\textrm{e}}re equation for a long time, we proved the convergence of this method until now in this paper. The theoretical convergence was established by using the regularity theory of the solution, the inverse inequality, the sampling inequality, and the approximation property of the radial basis functions trial spaces. However, the convergence rate is currently not yet optimal and deserve refinement.

References

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