Estimates of constants in error estimates for H^2 conforming finite elements for regularized nonlinear elliptic geometric evolution equations and question of optimality

1. Introduction

This paper extends the finite element error analysis for regularized geometric evolution equations considered in [13]. We are concerned with a finite element approximation formulated in Feng, Neilan, and Prohl [6] for the following family of equations

(1.1)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \begin{matrix} div ⎛ ⎜ ⎜ ⎝ \frac{\nabla u^{ε}}{\sqrt{| \nabla u^{ε} |^{2} + ε^{2}}} ⎞ ⎟ ⎟ ⎠ - η (\sqrt{| \nabla u^{ε} |^{2} + ε^{2}}) & = 0 & in Ω, u^{ε} & = 0 & on \partial Ω \end{matrix} \end{matrix}

with

(1.2)

η (r) := σ r^{α}

which was introduced in Huisken and Ilmanen [9] and Schulze [14] in order to approximate weak solutions of the inverse mean curvature flow and the flow by powers of the mean curvature, respectively. Before we formulate the issue under consideration in detail, we describe the equations more precisely. The parameter $ε$ is a positive regularization parameter which prevents the equations from becoming singular. We assume that $Ω \subset R^{3}$ is an open, bounded domain with a smooth boundary of positive mean curvature. The equation in case $ε = 0$ models (assuming sufficient regularity) level set motion (in the sense that the evolving surface at time $t \geq 0$ is given as the $t$ -level set of $u^{0}$ ) with normal speed given by a power of the mean curvature, so that we have in the case $σ = 1$ and $α = 1$ inverse mean curvature flow and in the case $σ = - 1$ and $α = - 1 / k$ , $k \geq 1$ , a contracting flow by a power $k$ of the mean curvature (this includes mean curvature flow for $k = 1$ ). Motivated by these applications for the rest of the paper we will always stay in these two cases for the regimes of $σ$ and $α$ . In the case $σ = - 1$ and $α = - 1 / k$ , $k \geq 1$ , equation (1.1) has smooth solutions for each sufficiently small $ε > 0$ , cf. [14]. Assuming that $u^{ε}$ converges to a continuous $u$ as $ε \to 0$ which satisfies equation (1.1) with $ε = 0$ in a viscosity sense (such a convergence is available in the case of the flow by a power of the mean curvature [14]), then, when aiming to solve equation (1.1) computationally with $ε = 0$ , it is tempting and reasonable to circumvent the possible singularity of the equation by approximating it with (1.1) for $ε > 0$ small and fixed. We remark that in the case $σ = α = 1$ the model equation (1.1) is a simplified case of the approximating problems for a weak formulation of inverse mean curvature flow in [9]. Therein the weak solutions of the inverse mean curvature flow are approximated by problems of type (1.1) in which in addition to $ε$ also the domain and the boundary values serve as a parameter in the approximation process and a certain minimization property is required in order to define weak solutions leading to a more complicated setting. Numerical error bounds which take these general approximating problems into account are not content of this paper, instead we focus on the simplified (not realistic in the case of the parameters chosen within the inverse mean curvature flow regime) case given by (1.1) in combination with $ε$ as the only parameter at first. In [6] from where we take the numerical scheme also numerical analysis only for this case is addressed. Therein the authors report about relevance of analyzing constants in FE error estimates (especially in the most general inverse mean curvature flow setting) and about some numerical experiences in special cases concerning the constants in error estimates. While the authors therein expect a polynomial growth of the constants in the inverse regularization parameter, our own experimental work [11] suggests a stronger growth which is in accordance with the exponential bound in the inverse regularization parameter from [13] and the present paper. Furthermore, in the final section of the present paper we give strong indication that our bound is optimal.

Let us explain the background and contribution of our paper. The work [6] shows $W^{1, p}$ - and $L^{p}$ -error bounds being of type of a product of a constant which depends in an unknown way on $ε$ and the usually expected powers of the discretization parameter. In previous work [13] we were able to derive an estimate of these constants in form of an upper bound $e^{P (1 / ε)}$ provided the finite elements are piecewise cubic, globally $H^{1}$ conform, the space dimension is $n + 1 = 3$ and by using an appropriate boundary approximation. ¹¹1Here and in the sequel $P$ applied to a list of positive arguments $l_{1}, . . ., l_{m}$ , $m \in N$ , stands for an expression which depends at most polynomial on $max {l_{i}, 1 / l_{i}}$ , $i = 1, \dots, m$ , hence to illustrate the notation in this example case we especially have

(1.3)

P (1 / ε) \leq \frac{c}{ε^{m}}

for small

ε

where

c, m > 0

are fixed constants which do not depend on

ε

and which may vary from line to line where the expression

P (1 / ε)

appears. The contribution of this paper is twofold. First, we extend the results from [13] to an

H^{2}

–conforming approach and second, we present a strong indication that one can not expect a better dependence of the constants in the numerical estimates on the inverse regularization parameter

ε

than exponential. Note that

H^{2}

–conformity is, at least in the two-dimensional case, a well-established approach; cf. the Argyris element or the Clough-Tocher element for the biharmonic equation; we remark, nevertheless also nonconforming approaches have shown their relevance in numerical computations. The resulting error estimates in correspondingly higher order Sobolev norms which in turn lead to

C^{1}

error estimates via embedding theorems are very reasonable from the view point of modeling evolving level sets. Evolving level sets cannot be detected from lower order error estimates. Explicit dependencies of the constants on the regularization parameter are needed when putting discretization error estimates together with pure regularization error estimates to a full error estimate. Estimates for the regularization errors can be found in [12] for the flow by powers of the mean curvature and in [11] in a crucially simplified rotationally symmetric case for the inverse mean curvature flow, see also [13] for a full error estimate in the

C^{0}

norm in certain cases. Note that

C^{0}

is from the view point of viscosity solutions a natural norm and that our higher order FE error estimates contribute usefully only to a full error estimate if the regularization error estimate (and the regularity of the solution) is of sufficiently high order; but they are in any case of own interest.

The quantitative convergence results for approximations of geometric evolution equations (with respect to the result itself and rather less concerning the methods and the technical level) developed in this paper as well as previous papers [11, 12, 13] of the authors are inspired by the works Crandall and Lions [3] and Deckelnick [4], the paper [3] shows convergence of a difference scheme to parabolic level set mean curvature flow, and [4] shows bounds for constants in error estimates (and in the sequel full error estimates) for a scheme based on [3] which are polynomial in the inverse regularization parameter. While in both references finite difference schemes for a parabolic problem are used, here finite element approximations for an elliptic problem are considered. Clearly, it is of interest to have also here convergence with rates available at which roughly said (note that for the precise statement the norms and limit processes have to be specified in details) our own contributions solely address the rates aspect while the convergence was already known before. The elliptic formulation has the advantage that even for nonlinear speeds in the mean curvature the equation is in divergence form and the nonlinearity arising from the velocity appears only in lower order terms. On the other hand parabolic problems are sometimes more convenient for numerical methods. An effect of this kind seems to appear here in the sense that for our special elliptic problem the dependence of constants on the regularization parameter (compared to the analogous issue in [4]) is of more implicit type, which we handle with a different method in comparison to [4], namely, by using techniques from [13]. See the parabolic versus elliptic discussion in Section 5.

Our approach uses $L^{\infty}$ –estimates due to the proof of the Alexandrov weak maximum principle [8, Thm. 9.1] which are available for the linearization of equation (1.1) in case of $H^{2}$ –conforming finite elements. While in that cited theorem explicit dependencies (between assumed bounds on the coefficients and the constant in the a priori estimate) are not formulated, its proof allows to work out these more explicitly yielding an exponential relation. For additional references about numerical analysis for geometric evolution equations we refer to the variety of references in [11, 10, 7] as well as to these papers themselves.

In this final section we discuss some practical experiences concerning the issue of polynomial and exponential rates. Numerical examples for a rotationally symmetric setting (the symmetry avoids the singularity of the equation in the computational domain) using $H^{1}$ –conforming finite elements are presented in [6]. Thereby the authors apply a moderate coupling of the regularization and discretization parameter $ε$ and $h$ of type $h \approx ε^{2}$ . Under this polynomial coupling the numerical scheme seems to work well in the considered numerical examples; for further details, see [6]. For a more general case (without symmetry and with singularity in the computational domain) the observation from [11] is that numerical computations with small $ε$ are difficult at all which makes it at least plausible that the bound could be worse than of moderate polynomial type in the case of $H^{1}$ –conforming finite elements, e.g. even exponential.

Note that the method presented here generalizes to several other situations and equations which inherit certain parameters for which a certain asymptotic is assumed.

2. Main result

We adopt the setting from [13]. Throughout the paper we use for a bounded domain $Ω \subset R^{n + 1}$ the usual notation for Lebesgue spaces $L^{p} (Ω)$ , $1 \leq p \leq \infty$ , and Sobolev spaces $W^{m, p} (Ω)$ , $0 \leq m < \infty$ , $1 \leq p \leq \infty$ . For $p = 2$ we write $H^{m} (Ω) = W^{m, 2} (Ω)$ . By $H_{0}^{1} (Ω)$ we denote the closure of the smooth functions being compactly supported in $Ω$ in the $H^{1}$ –norm. We denote generic constants in estimates usually by $c$ and use the summation convention that we sum over repeated indices. By $| \cdot |$ we denote the Euclidean norm, and by $| \cdot |_{1}$ the $1$ -norm. The domain of the level set function is considered to be $n + 1 = 3$ dimensional, so that level sets are in case of sufficient regularity $n = 2$ dimensional surfaces. Let

(2.1)

{T_{h} : 0 < h < h_{0}}

be a family of shape-regular and uniform triangulations of $Ω$ , $h$ the mesh size of $T_{h}$ and $h_{0} = h_{0} (Ω) > 0$ small, where we allow boundary elements to be curved and define

(2.2)

Ω^{h} := \cup_{T \in T_{h}} T;

since $Ω$ might lack convexity, we have not in general $Ω^{h} \subset ¯ Ω$ . We will specify the triangulation concerning their boundary approximation in the following two assumptions and will already here stipulate that we will consider piecewise polynomial functions with polynomial degree at most $deg \geq 3$ , $deg \in N$ .

Assumption 2.1.

For $h > 0$ we assume the existence of the following $H^{2}$ -conforming finite element space:

(2.3)

V_{h} :=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} w \in C^{1} ({¯ Ω}^{h}) ∣ ∣ ∣ \begin{matrix} for all T \in T_{h} : w |_{T} % polynomial of degree \leq d e g (up to the transformation in case T is a boundary cell), w_{| \partial Ω^{h}} = 0. \end{matrix} \end{matrix} ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ .

Hereby, we allow for the finite elements restricted to the boundary cells to be accordingly transformed polynomials, see Appendix A.

The corresponding space when dropping the requirement $w_{| \partial Ω^{h}} = 0$ in (2.3) is denoted by ${~ V}_{h}$ .

Assumption 2.2.

For $0 < h \leq h_{0}$

we assume that there exists an interpolation operator

I_{h} : H^{d e g} (Ω^{h}) \to {~ V}_{h}

such that for

1 \leq p \leq \infty

(2.4)

∥ u - I_{h} u ∥_{W^{m, p} (Ω^{h})} \leq c h^{d e g - m} ∥ u ∥_{W^{d e g, p} (Ω^{h})} \forall u \in H^{deg} (Ω^{h}), m = 1, 2.

Furthermore, there is a constant $0 < ~ c := ~ c (Ω)$ so that

(2.5)

\partial Ω^{h} \subset Ω_{~ c h^{˜ d e g}} ∖ Ω_{- ~ c h^{˜ d e g}} .

where we use the notation (2.6) and assume that $˜ d e g \leq \frac{d e g}{2}$ , $˜ d e g \in N$ .

Note that the $˜ d e g \leq d e g / 2$ , $˜ d e g \in N$ , inequality is assumed for simplification and is to avoid a later consideration of two different cases and that the excluded range for $˜ d e g$ could be treated analogously but is not that interesting since it corresponds to a rather high boundary approximation.

We denote the set of nodes of $T_{h}$ by $N_{h}$ . We recall that a continuous piecewise polynomial function whose derivatives are continuous along faces is an element in $H^{2} (Ω)$ , cf. [1, Thm. 5.2]. Since the curved elements at the boundary can be treated in the weak formulation of the discrete problem analogously as if they were exact tetrahedra we will refer in the following in our notation to these elements as to the usual tetrahedra. Let $d : R^{3} \to R$ be the signed distance function of $\partial Ω$ where the sign convention is so that $d$ is negative inside $Ω$ and nonnegative outside $Ω$ . Let $δ_{0} = δ_{0} (Ω) > 0$ be small and define for $0 < δ < δ_{0}$ that

(2.6)

Ω_{δ} := {x \in R^{3} | d (x) < δ}

then we have

(2.7)

\partial Ω_{δ} \in C^{\infty}, ∥ \partial Ω_{δ} ∥_{C^{2}} \leq c (Ω) ∥ \partial Ω ∥_{C^{2}} .

Let $h_{0} > 0$ be chosen such that

(2.8)

Ω^{h} \subset Ω_{δ_{0}} for all 0 < h \leq h_{0} .

We extend $u^{ε}$ to a function in $C^{m} (Ω_{δ_{0}})$ with $m \in N$ sufficiently large, denote the extension again $u^{ε}$ , and assume that

(2.9)

∥ u^{ε} ∥_{C^{m} (Ω_{δ_{0}})} \leq c ∥ u^{ε} ∥_{C^{m} (¯ Ω)} .

Furthermore, when (tacitly) extending functions $v_{h} \in V_{h}$ to $R^{n + 1}$ by zero we denote the extended function again by $v_{h}$ .

We formulate our main result.

Theorem 2.3.

Let $μ \geq 2$ and $ν := \frac{3}{μ} - \frac{7}{2} + \frac{4}{3} ˜ d e g$ . Then choosing $δ$ such that

(2.10)

{\begin{matrix} ν > δ > 2, & if μ > 3, ν > δ > \frac{3}{μ} - \frac{1}{2}, & if 2 \leq μ \leq 3 \end{matrix}

and setting

(2.11)

ρ = e^{ε^{- γ}} h^{δ}

with $γ > 0$ suitable constant with can be calculated explicitly the following holds: For every $0 < ε < ε_{0}$ and $0 < h \leq h_{0}$ the equation

(2.12)

\int_{Ω^{h}} \frac{⟨ D u_{h}^{ε}, D φ_{h} ⟩}{\sqrt{ε^{2} + | D u_{h}^{ε} |^{2}}} = - \int_{Ω^{h}} η (\sqrt{} ε^{2} + | D u_{h}^{ε} |^{2}) φ_{h} \forall φ_{h} \in V_{h},

has a unique solution $u_{h}^{ε}$ in

(2.13)

{¯ B}_{ρ}^{h} := {w_{h} \in V_{h} : ∥ w_{h} - u^{ε} ∥_{H^{2, μ} (Ω^{h})} \leq ρ} .

Remark 2.4.

Condition (2.10) implies a lower bound for $˜ d e g$ , namely $˜ d e g > \frac{3}{4} (\frac{11}{2} - \frac{3}{μ})$ for $μ > 3$ and $˜ d e g > \frac{9}{4}$ for $2 \leq μ \leq 3$ . Furthermore, we have $˜ d e g \leq \frac{d e g}{2}$ , see the discussion below Assumption 2.2.

Throughout the paper we assume $μ \geq 2$ which will be used in (4.22).

Example 2.5.

For $˜ d e g = 3$ , $d e g = 9$ (see for a realization Section A), and $μ = 2$ the condition for $δ$ reads as follows:

(2.14)

2 > δ > 1.

In comparison with our previous result [13, Thm. 1.2] there are two main differences: In [13]

we used $H^{1}$ instead of $H^{2}$ –conforming finite elements,
we estimated $W^{1, μ}$ - instead of $W^{2, μ}$ –norms.

Furthermore, note that the validity of the assumptions in Theorem 2.3—exactly seen—do not imply that the assumptions of the paper [13] are satisfied because we fixed therein the concrete space of piecewise polynomial functions (while here we keep the space more abstract), nevertheless, clearly, an easy observation shows that the former conclusion holds for the present case as well.

The remaining part of the paper deals with the proof of Theorem 2.3.

3. Discrete $W^{2, μ}$ -estimates with explicit constants

Here, we formulate some auxiliary estimates for a class of linear equations which includes in particular the linearization of (1.1); we proceed analogously to [13].

The linearized operator $L_{ε} := L_{ε} (u^{ε}) : H_{0}^{1} (Ω) \to H^{- 1} (Ω)$ for (1.1) in a solution $u^{ε}$ is given by

(3.1)

L u := D_{i} (a^{i j} D_{j} u) + c^{i} D_{i} u + d u

in $Ω$ with coefficients given as follows (such a calculation can be found in [13]): We define for $ε > 0$ and $z \in R^{n + 1}$

(3.2)

| z |_{ε} := f_{ε} (z) := \sqrt{| z |^{2} + ε^{2}}

and denote derivatives of $f_{ε}$ with respect to $z^{i}$ by $D_{z^{i}} f_{ε}$ , i.e. there holds

(3.3)

D_{z^{i}} f_{ε} (z) = \frac{z_{i}}{| z |_{ε}}, D_{z^{i}} D_{z^{j}} f_{ε} (z) = \frac{δ_{i j}}{| z |_{ε}} - \frac{z_{i} z_{j}}{| z |_{ε}^{3}};

with these notations we set

(3.4)

a^{i j} := - D_{z^{i}} D_{z^{j}} f_{ε} (D u^{ε}), c^{i} := η^{'} (| D u^{ε} |) D_{z^{i}} f_{ε} (D u^{ε}) % and d := 0.

Furthermore, we introduce the equation

(3.5)

L u = g + D_{i} f^{i} .

with $f^{i} \in W^{1, p} (Ω)$ , $1 \leq i \leq n + 1$ , and $g$ in $L^{p} (Ω)$ , $p \geq 1$ . The special formal structure of the right-hand side is chosen as in [13] where when working with $H^{1}$ conforming elements one only has $f^{i} \in L^{p} (Ω)$ .

For convenience we recall the following well-known inverse estimate [2, Sec. 4.5] which will be used without mentioning it each time.

Lemma 3.1.

For $1 \leq p, q \leq \infty$ and $0 \leq m \leq l \leq 2$ there exists a constant $c > 0$ such that

(3.6)

∥ v_{h} ∥_{W^{l, p} (Ω^{h})} \leq c h^{m - l + min (0, \frac{n}{p} - \frac{n}{q})} ∥ v_{h} ∥_{W^{m, q} (Ω^{h})}

for all $v_{h} \in V_{h}$ .

Using standard elliptic regularity theory it is shown in [13, Sec. 2] that the solution $u^{ε}$ of (1.1) is smooth and satisfies the estimate

(3.7)

∥ u^{ε} ∥_{H^{m} (Ω)} = P (1 / ε)

for all $m \in N$ .

Lemma 3.2.

Let $δ_{0}$ be given as in Section 2 and $h_{0} > 0$ such that (2.8) is satisfied. We allow unless specified concretely that

(3.8)

{\begin{matrix} ~ Ω & = Ω^{h} with 0 < h < h_{0} or ~ Ω & = Ω_{δ_{0}} \end{matrix}

and assume $D_{i} f^{i} + g \in L^{n + 1} (~ Ω)$ .

(i) There exists a unique solution $u \in H^{2} (~ Ω) \cap H_{0}^{1} (~ Ω)$ of (3.5) with $L = L_{ϵ}$ satisfying

(3.9)

{∥ u ∥}_{H^{2} (~ Ω)}

\leq e^{P (1 / ε)} ∥ D_{i} f^{i} + g ∥_{L^{n + 1} (~ Ω)} .

(ii) Furthermore, assuming $~ Ω = Ω^{h}$ in (i) we have the best-approximation property

(3.10)

{∥ u - u_{h} ∥}_{H^{1} (Ω^{h})} \leq e^{P (1 / ε)} inf v_{h} \in V_{h} {∥ u - v_{h} ∥}_{H^{1} (Ω^{h})} .

Proof.

(i) As an intermediate step we prove

(3.11)

∥ D u ∥_{L^{2} (~ Ω)}

\leq e^{P (1 / ε)} ∥ D_{i} f^{i} + g ∥_{L^{n + 1} (~ Ω)} .

This estimate follows from [13, Lem. 7.1] replacing [13, Thm. 6.2] therein by the $L^{\infty}$ –estimate in Theorem B.1. Thereby, $D_{i} f^{i} + g$ from our setting plays the role of $g$ in [13, Lem. 7.1], note that the two situations differ since we assume now higher regularity for $f^{i}$ . Estimate (3.9) is a straightforward calculation by combining the standard proof for higher regularity with the estimate (3.11).

(ii) The proof follows exactly the lines in [13, estimate (7.12)] using (3.9) instead of [13, (7.10)]; we remark that in the latter reference cubic elements are used, however, since here we have ansatz functions of higher polynomial degree and a boundary approximation as given in (2.5), the situation here is even at least as convenient as before and allows the application of the former arguments. ∎

Theorem 3.3.

Let $u$ be the unique solution of (3.5) in $Ω^{h}$ where $L = L_{ϵ}$ . Then, there is $h_{0} > 0$ so that for

(3.12)

0 < h \leq h_{0}

there exists a unique finite element solution $u_{h} \in V_{h}$ of (2.12) in $Ω^{h}$ satisfying

(3.13)

∥ u_{h} ∥_{H^{2} (Ω^{h})} \leq e^{P (1 / ε)} ∥ D_{i} f^{i} + g ∥_{L^{n + 1} (Ω^{h})} .

Proof.

We have

(3.14)	${∥ u - u_{h} ∥}_{H^{2} (Ω^{h})}$	$\leq {∥ u - I_{h} u ∥}_{H^{2} (Ω^{h})} + {∥ u_{h} - I_{h} u ∥}_{H^{2} (Ω^{h})}$
		$\leq {∥ u - I_{h} u ∥}_{H^{2} (Ω^{h})} + C h^{- 1} {∥ u_{h} - I_{h} u ∥}_{H^{1} (Ω^{h})}$
		$\leq {∥ u - I_{h} u ∥}_{H^{2} (Ω^{h})} + C h^{- 1} ({∥ u - I_{h} u ∥}_{H^{1} (Ω^{h})} + {∥ u - u_{h} ∥}_{H^{1} (Ω^{h})})$
		$\leq {∥ u - I_{h} u ∥}_{H^{2} (Ω^{h})} + C {∥ u ∥}_{H^{2} (Ω^{h})} + C h^{- 1} {∥ u - u_{h} ∥}_{H^{1} (Ω^{h})};$

here, $I_{h}$ is the interpolation operator introduced in Assumption 2.2. Since by (3.10)

(3.15)

we have by interpolation estimate (2.4) that

(3.16)

{∥ u - u_{h} ∥}_{H^{2} (Ω^{h})}

\leq e^{P (1 / ε)} {∥ ∥ D_{i} f^{i} + g ∥ ∥}_{L^{n + 1} (Ω^{h})}^{i}

and the claim follows from the triangle inequality and stability estimate (3.9).

∎

Note that we will apply Banach’s fixed point theorem in $W^{2, μ} (Ω^{h})$ –balls but go via $H^{2} (Ω^{h})$ –norms for the reason to work out the constants explicitly, this is analogous to [13] where we apply Banach’s fixed point theorem in $W^{1, μ} (Ω^{h})$ –balls.

4. Banach’s fixed point theorem in $W^{2, μ}$ balls with radii given explicitly in terms of $h$ and $ε$

We prove Theorem 2.3.

(i) We present the general outline of the proof which follows the proofs in [6, Sec. 2] and [13, Thm. 4.2] with the following differences. The novelties which we implement here in the proof are the higher order Sobolev norms and explicit constants. Both is new compared with [6] and the first (and hence the combination of both) also compared with [13]. The higher regularity in the present paper allows to differentiate second order derivative expressions explicitly which makes the estimates different from the latter reference. Existence and uniqueness of a solution $u_{h}^{ε}$ of (2.12) will be shown by identification of this solution as the unique fixed point of a map $T : V_{h} \to V_{h}$ in ${¯ B}_{ρ}^{h}$ , cf. (2.13), which will be defined in (4.4). Uniqueness and existence of the fixed point follows by Banach’s fixed point theorem. Therefore, we will check the standard assumptions of the fixed point theorem in a quantitative way with respect to constants. We use the following selection of three sufficient conditions:

(4.1)

{¯ B}_{ρ}^{h} \neq \emptyset,

recalling $2 \leq μ \leq 3$ condition

(4.2)

∥ T w_{h} - T v_{h} ∥_{W^{2, μ} (Ω^{h})} \leq c h^{η} ∥ w_{h} - v_{h} ∥_{W^{2, μ} (Ω^{h})} \forall w_{h}, v_{h} \in {¯ B}_{ρ}^{h}

with some $η > 0$ , and

(4.3)

T ({¯ B}_{ρ}^{h}) \subset {¯ B}_{ρ}^{h} .

Hereby, we define $T : V_{h} \to V_{h}$ by

(4.4)

L_{ε} (w_{h} - T w_{h}) = Φ_{ε} (w_{h}), w_{h} \in V_{h}

with the operator $Φ_{ε}$ given by

(4.5)

Φ_{ε} : H_{0}^{1} (Ω^{h}) \to H^{- 1} (Ω^{h}), Φ_{ε} (v) := - D_{i} (\frac{D_{i} v}{| D v |_{ε}}) + η (| D v |_{ε}) .

(ii) Condition (4.1) follows directly from (2.4).

(iii) Here, an estimate for $∥ T (v_{h} - w_{h}) ∥_{W^{2, μ} (Ω^{h})}$ with $μ \geq 2$ is shown. Let $v_{h}$ and $w_{h}$ be in ${¯ B}_{ρ}^{h}$ , $ξ_{h} = v_{h} - w_{h}$ , $α (t) = w_{h} + t ξ_{h}$ , $0 \leq t \leq 1$ . In view of (4.4) we have

(4.6)

L_{ε} (T v_{h} - T w_{h}) = L_{ε} ξ_{h} + Φ_{ε} (w_{h}) - Φ_{ε} (v_{h}) .

Recalling the convention that when $η$ and $f_{ε}$ have no arguments it is meant $η = η (| D u^{ε} |_{ε})$ and $f_{ε} = f_{ε} (u^{ε})$ the right-hand side of (4.6) is of the form $D_{i} f^{i} + g \in L^{n + 1}$ with (see [13])

(4.7)		$f^{i}$	$= D_{z^{i}} f_{ε} (D v_{h}) - D_{z^{i}} f_{ε} (D w_{h}) - D_{z^{i}} D_{z^{j}} f_{ε} D_{j} ξ_{h}$
(4.7)			$= \int_{0}^{1} (D_{z^{j}} D_{z^{i}} f_{ε} (D α (t)) - D_{z^{i}} D_{z^{j}} f_{ε}) D_{j} ξ_{h}$

and (using the convention that $η^{'}$ denotes the derivative of simply the function $r \mapsto η (r)$ and that the argument if omitted is understood to be $| D u^{ε} |_{ε}$ ) we have

(4.8)		$g$	$= η^{'} D_{z^{j}} f_{ε} D_{j} ξ_{h} + η (f_{ε} (D w_{h})) - η (f_{ε} (D v_{h}))$
(4.8)			$= \int_{0}^{1} (η^{'} D_{z^{j}} f_{ε} - η^{'} (f_{ε} (D α (t))) D_{z^{j}} f_{ε} (D α (t))) D_{j} ξ_{h} .$

Since the finite element space is $H^{2}$ –conforming we may rewrite $D_{i} f^{i}$ by performing the differentiation and get by using the abbreviation

(4.9)

G (t) = D_{z^{r}} D_{z^{i}} f_{ε} (D α (t)) D_{i} D_{r} α (t)

that

(4.10)	$D_{i} f^{i}$	$= D_{z^{r}} D_{z^{i}} f_{ε} (D α (1)) D_{i} D_{r} α (1) - D_{z^{r}} D_{z^{i}} f_{ε} (D α (0)) D_{i} D_{r} α (0)$
		$- D_{z^{r}} D_{z^{i}} D_{z^{j}} f_{ε} D_{i} D_{r} u^{ε} D_{j} ξ_{h} - D_{z^{i}} D_{z^{j}} f_{ε} D_{i} D_{j} ξ_{h}$
		$= \int_{0}^{1} \frac{d}{d t} G (t) - D_{z^{r}} D_{z^{i}} D_{z^{j}} f_{ε} D_{i} D_{r} u^{ε} D_{j} ξ_{h} - D_{z^{i}} D_{z^{j}} f_{ε} D_{i} D_{j} ξ_{h} .$

There holds

(4.11)		$\frac{d}{d t} G (t)$	$= D_{z^{m}} D_{z^{r}} D_{z^{i}} f_{ε} (D α (t)) D_{m} ξ_{h} D_{i} D_{r} α (t)$
(4.11)			$+ D_{z^{r}} D_{z^{i}} f_{ε} (D α (t)) D_{i} D_{r} ξ_{h} .$

Now we rewrite $\frac{d}{d t} G (t)$ by using the identity

(4.12)

A B C = (A - a) (B - b) C + (A - a) b C + a (B - b) C + a b C

for real numbers $A, B, C, a, b$ , i.e. we have

(4.13)	$\frac{d}{d t} G (t)$	$= (D_{z^{m}} D_{z^{r}} D_{z^{i}} f_{ε} (D α (t)) - D_{z^{m}} D_{z^{r}} D_{z^{i}} f_{ε} (D u^{ε})) D_{m} ξ_{h} (D_{i} D_{r} α (t) - D_{i} D_{r} u^{ε})$
		$+ (D_{z^{m}} D_{z^{r}} D_{z^{i}} f_{ε} (D α (t)) - D_{z^{m}} D_{z^{r}} D_{z^{i}} f_{ε} (D u^{ε})) D_{m} ξ_{h} D_{i} D_{r} u^{ε}$
		$+ D_{z^{m}} D_{z^{r}} D_{z^{i}} f_{ε} (D u^{ε}) D_{m} ξ_{h} (D_{i} D_{r} α (t) - D_{i} D_{r} u^{ε})$
		$+ D_{z^{m}} D_{z^{r}} D_{z^{i}} f_{ε} (D u^{ε}) D_{m} ξ_{h} D_{i} D_{r} u^{ε}$
		$+ (D_{z^{r}} D_{z^{i}} f_{ε} (D α (t)) - D_{z^{r}} D_{z^{i}} f_{ε} (D u^{ε})) D_{i} D_{r} ξ_{h} + D_{z^{r}} D_{z^{i}} f_{ε} (D u^{ε}) D_{i} D_{r} ξ_{h}$

Due to cancellations we can write

(4.14)	$D_{i} f^{i}$	$= \int_{0}^{1} [(D_{z^{m}} D_{z^{r}} D_{z^{i}} f_{ε} (D α (t)) - D_{z^{m}} D_{z^{r}} D_{z^{i}} f_{ε} (D u^{ε})) D_{m} ξ_{h} (D_{i} D_{r} α (t) - D_{i} D_{r} u^{ε})$
		$+ (D_{z^{m}} D_{z^{r}} D_{z^{i}} f_{ε} (D α (t)) - D_{z^{m}} D_{z^{r}} D_{z^{i}} f_{ε} (D u^{ε})) D_{m} ξ_{h} D_{i} D_{r} u^{ε}$
		$+ D_{z^{m}} D_{z^{r}} D_{z^{i}} f_{ε} (D u^{ε}) D_{m} ξ_{h} (D_{i} D_{r} α (t) - D_{i} D_{r} u^{ε})$
		$+ (D_{z^{r}} D_{z^{i}} f_{ε} (D α (t)) - D_{z^{r}} D_{z^{i}} f_{ε} (D u^{ε})) D_{i} D_{r} ξ_{h}] d t .$

We will apply Theorem 3.3 with the right-hand side given by (4.6), or more explicitly expressed by using (4.14) and (4.8) and the estimates below. The needed $L^{n + 1} (Ω^{h})$ norm of that right-hand side will be related for that purpose to available norms of the expressions on the right-hand side by using the interpolation estimate (2.4) and switching between discrete norms by applying inverse estimates. We then arrive at

(4.15)	$∥ D w_{h} - D u^{ε} ∥_{L^{\infty} (Ω^{h})}$	$\leq ∥ D w_{h} - D I_{h} u^{ε} ∥_{L^{\infty} (Ω^{h})} + ∥ D I_{h} u^{ε} - D u^{ε} ∥_{L^{\infty} (Ω^{h})}$
		$\leq c h^{1 - \frac{n + 1}{μ}} (∥ D w_{h} - D u^{ε} ∥_{W^{1, μ} (Ω^{h})}$
		$+ ∥ D u^{ε} - D I_{h} u^{ε} ∥_{W^{1, μ} (Ω^{h})}) + c h^{d e g - 1} ∥ u^{ε} ∥_{C^{d e g} ({¯ Ω}^{h})}$
		$\leq c h^{1 - \frac{n + 1}{μ}} (ρ + h^{d e g - 2} ∥ u^{ε} ∥_{C^{d e g} ({¯ Ω}^{h})}) + c h^{d e g - 1} ∥ u^{ε} ∥_{C^{d e g} ({¯ Ω}^{h})} .$

Analogously, we have

(4.16)	$∥ D^{2} w_{h} - D^{2} u^{ε} ∥_{L^{\infty} (Ω^{h})}$	$\leq ∥ D^{2} w_{h} - D^{2} I_{h} u^{ε} ∥_{L^{\infty} (Ω^{h})} + ∥ D^{2} I_{h} u^{ε} - D^{2} u^{ε} ∥_{L^{\infty} (Ω^{h})}$
		$\leq c h^{- \frac{n + 1}{μ}} (∥ D^{2} w_{h} - D^{2} u^{ε} ∥_{L^{μ} (Ω^{h})}$
		$+ ∥ D^{2} u^{ε} - D^{2} I_{h} u^{ε} ∥_{L^{μ} (Ω^{h})}) + c h^{d e g - 2} ∥ u^{ε} ∥_{C^{d e g} ({¯ Ω}^{h})}$
		$\leq c h^{- \frac{n + 1}{μ}} (ρ + h^{d e g - 2} ∥ u^{ε} ∥_{C^{d e g} ({¯ Ω}^{h})}) + c h^{d e g - 2} ∥ u^{ε} ∥_{C^{d e g} ({¯ Ω}^{h})} .$

Clearly, since $ρ$ plays the role of the radius of the ball ${¯ B}_{ρ}^{h}$ in which we confirm the assumptions of Banach’s fixed point theorem (and we are interested in the asymptotic regime of the parameter values) it will naturally have a small value so that we may assume w.l.o.g. its boundedness by a moderate constant, e.g. $ρ < 1$ , which will be used for the following tacitly (and will vanish in generic constants). We estimate by using (4.14), (4.15) and (4.16) as follows

(4.17)			$∥ D_{i} f^{i} ∥_{L^{n + 1} (Ω^{h})} \leq e^{P (1 / ε)} (∥ D w_{h} - D u^{ε} ∥_{L^{\infty} (Ω^{h})} + ∥ D v_{h} - D u^{ε} ∥_{L^{\infty} (Ω^{h})})$
			$∥ D ξ_{h} ∥_{L^{n + 1} (Ω^{h})} (∥ D^{2} w_{h} - D^{2} u^{ε} ∥_{L^{\infty} (Ω^{h})} + ∥ D^{2} v_{h} - D^{2} u^{ε} ∥_{L^{\infty} (Ω^{h})})$
			$+ e^{P (1 / ε)} (∥ D w_{h} - D u^{ε} ∥_{L^{\infty} (Ω^{h})} + ∥ D v_{h} - D u^{ε} ∥_{L^{\infty} (Ω^{h})}) ∥ D ξ_{h} ∥_{L^{n + 1} (Ω^{h})}$
			$e^{P (1 / ε)} (∥ D^{2} w_{h} - D^{2} u^{ε} ∥_{L^{\infty} (Ω^{h})} + ∥ D^{2} v_{h} - D^{2} u^{ε} ∥_{L^{\infty} (Ω^{h})}) ∥ D ξ_{h} ∥_{L^{n + 1} (Ω^{h})}$
			$+ e^{P (1 / ε)} (∥ D w_{h} - D u^{ε} ∥_{L^{\infty} (Ω^{h})} + ∥ D v_{h} - D u^{ε} ∥_{L^{\infty} (Ω^{h})}) ∥ D^{2} ξ_{h} ∥_{L^{n + 1} (Ω^{h})} .$

Furthermore, we have

(4.18)

∥ g ∥_{L^{n + 1} (Ω^{h})} \leq e^{P (1 / ε)} (∥ D w_{h} - D u^{ε} ∥_{L^{\infty} (Ω^{h})} + ∥ D u^{ε} - D v_{h} ∥_{L^{\infty} (Ω^{h})}) ∥ D ξ_{h} ∥_{L^{n + 1} (Ω^{h})} .

We estimate the right-hand sides of (4.17) and (4.18) further from above in terms of $ρ$ , $h$ and $ε$ which will we done by relating the appearing $L^{n + 1} (Ω^{h})$ norms to $L^{μ}$ norms and by using the estimate

(4.19)

∥ D^{m} w_{h} - D^{m} u^{ε} ∥_{L^{\infty} (Ω^{h})}

\leq ∥ D^{m} w_{h} - D^{m} I_{h} u

Estimates of constants in error estimates for H^2 conforming finite elements for regularized nonlinear elliptic geometric evolution equations and question of optimality

Authors

Convergence of a regularized finite element discretization of the two-dimensional Monge-Ampère equation

A convergent finite element algorithm for generalized mean curvature flows of closed surfaces

Robust discretization and solvers for elliptic optimal control problems with energy regularization

A convergent finite element algorithm for mean curvature flow in higher codimension

Rethinking Exponential Averaging of the Fisher

A converging finite element scheme for motion by curvature of a network with a triple junction

Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps

1. Introduction

2. Main result

Assumption 2.1.

Assumption 2.2.

Theorem 2.3.

Remark 2.4.

Example 2.5.

3. Discrete $W^{2, μ}$ -estimates with explicit constants

Lemma 3.1.

Lemma 3.2.

Proof.

Theorem 3.3.

Proof.

4. Banach’s fixed point theorem in $W^{2, μ}$ balls with radii given explicitly in terms of $h$ and $ε$

Estimates of constants in error estimates for H^2 conforming finite elements for regularized nonlinear elliptic geometric evolution equations and question of optimality

Authors

Related Research

Convergence of a regularized finite element discretization of the two-dimensional Monge-Ampère equation

A convergent finite element algorithm for generalized mean curvature flows of closed surfaces

Robust discretization and solvers for elliptic optimal control problems with energy regularization

A convergent finite element algorithm for mean curvature flow in higher codimension

Rethinking Exponential Averaging of the Fisher

A converging finite element scheme for motion by curvature of a network with a triple junction

Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps

This week in AI

1. Introduction

2. Main result

Assumption 2.1.

Assumption 2.2.

Theorem 2.3.

Remark 2.4.

Example 2.5.

3. Discrete W2,μ-estimates with explicit constants

Lemma 3.1.

Lemma 3.2.

Proof.

Theorem 3.3.

Proof.

4. Banach’s fixed point theorem in W2,μ balls with radii given explicitly in terms of h and ε

3. Discrete $W^{2, μ}$ -estimates with explicit constants

4. Banach’s fixed point theorem in $W^{2, μ}$ balls with radii given explicitly in terms of $h$ and $ε$