Some estimates of virtual element methods for fourth order problems

1.   Introduction

Virtual Element methods are designed on polygonal/polyhedral meshes, see [2]–[5]. It gives us the flexibility to use a wide range of meshes which is a great advantage especially in computational mechanics. And the computation cost is less than weak Galerkin finite element method, which can also employ general shape elements, see [17]–[14]. Virtual Element Methods for second order problems are well studied in [2]–[5], also the stability and error analysis for these methods are obtained. New techniques based on the shape regularity and discrete norm for virtual element functions are developed in [7]. Virtual Element Methods for fourth order problems are analyzed in [10,11], however, the stability and error analysis are not completed.

The motivations for this paper are: firstly, apply the techniques in [7] to higher order problems, getting the basic estimates; secondly, to improve the error analysis for biharmonic equation. If we modify the virtual element method slightly then the regularity requirement for right hand side function $f$ can be relaxed; thirdly, since the numerical solutions $u_h$ of virtual element methods can't be computed directly, to overcome this drawback, we present two ways to get the approximations of $u_h$ preserving the same convergence rate, and the computation of the approximations is much more efficient.

The paper is organized as follows: In Section 2, 2.1–2.3, the definition of two dimensional virtual element with the shape regularity is given. The projection in 2.2 is the same as in [11]. Compared with the definition of local virtual element space in [10,11], our method has the right hand side polynomial $q_v( = \Delta^2v) \in \mathbb{P}_k(D)$ not $\mathbb{P}_{k-4}(D)$ or $\mathbb{P}_{k-2}(D)$. This new definition provides more degrees of freedom which is necessary to denote the $L^2$ projection from virtual element function space to $\mathbb{P}_k(D)$. In Section 2.4, a semi-norm $|||\cdot|||_{k, D}$ similar as in [7] is presented. The local estimates for the projections $\Pi_{k, D}^\Delta$ and $\Pi_{k, D}^0$ are obtained. In Section 2.5–2.6, a piecewise $C^1$ polynomial $w$ depending only on the values on $\partial D$ is constructed and the local interpolation error is proved. In section 3, we get the error estimates between $u_h$, its approximations and the exact solution for biharmonic equation. In Section 4, we draw the conclusions.

2.   Local virtual element spaces in two dimensions

Let $D$ be a polygon in $\mathbb{R}^2$ with diameter $h_D$. For a nonnegative integer $k$, $\mathbb{P}_k$ is the space of polynomials of degree $\leq k$ and $\mathbb{P}_{-k} = \{0\}, \ k \geq 1.$ The space $\mathbb{P}_k(D)$ is the restriction of $\mathbb{P}_k$ to $D.$

The index $(r, s)$ related to the degree of $k\geq 2$ is defined by

A natural choice is $r = \max\{3, k\}$, however, $r$ can be greater. The set of edges of $D$ is denoted by $\mathcal{E}_D$ and $\mathbb{P}_k(e)$ is the restriction of $\mathbb{P}_k$ to $e\in \mathcal{E}_D.$ Then we define $\mathbb{P}_{r, s}(\partial D)$ as

2.1.   Shape regularity

Here the shape regularity assumptions are the same as in [7]. Let ${D}$ be the polygon with diameter $h_D$. Assume that

and

The center of $\mathfrak{B}$ is the star center of ${D}.$

The polygon in Figure 1 is an example of ${D}$, we denote by $\mathcal{T}_D$ the corresponding triangulation of D. We will use the notation $A \lesssim B$ to represent the inequality $A \leq (constant)B$, where the positive constant depends only on $k$ and the parameter $\rho$, and it increases with $k$ and $1/\rho$. The notation $A \approx B$ is equivalent to $A \lesssim B$ and $A \gtrsim B$.

Lemma 2.1. [6] Bramble-Hilbert Estimates. Conditions (1)-(2) imply that we have the following estimates:

for any $\xi\in H^{l+1}(D), \ l = 0, \cdots, k, \ and\ 0\leq m \leq l.$

Details can be found in [8], Lemma 4.3.8.

Lemma 2.2. [1] Sobolev Imbedding Theorem 4.12. From (1)-(2), we have:

Lemma 2.3. [16] The Generalized Poincar$\acute{\rm e}$ Inequality. From (1)-(2), suppose $h_D = 1$, we have:

Proof. The proof is similar as the one in [16], Section 1.1.6.

2.2.   The projection $\Pi_{k, D}^\Delta$

By the generalized poincar$\acute{\rm e}$ inequality from Lemma 2.3, the Sobolev space $H^2(D)$ is a Hilbert space with the inner product $(((\cdot)))$ denoted as:

where

for any $\xi, v \in H^{2}(D).$

The discrete operator $\Pi_{k, D}^\Delta: H^2(D)\rightarrow \mathbb{P}_k(D)$ is denoted with respect to $(((\cdot)))$ as:

in (7), let $q = 1,$ we have (8), let $q = x$ then $q = y$, we have (9), with the help of (8) and (9), the definition of $\Pi_{k, D}^\Delta$ in (7) can be simplified as (10),

from now on we will use (8)-(10) instead of (7).

On the domain $D$, with boundary $\partial D$, we denote by ${\bf n} = (n_1, n_2)$ the outward unit normal vector to $\partial D$, and by ${\bf t} = (t_1, t_2)$ the unit tangent vector in the counterclockwise ordering of the boundary. For $u\in H^2(D)$, we define

We then denote by $U_{nn}({\bf D}^2 u) : = \sum\limits_{i, j}u_{ij} n_i n_j$ the normal bending moment, by $U_{nt}({\bf D}^2 u) : = \sum\limits_{i, j}u_{ij} n_i t_j$ the twisting moment, and by $Q_{n}({\bf D}^2 u) : = \sum\limits_{i, j}\frac{\partial u_{ij}}{\partial x_i} n_j$ the normal shear force, and $U_{\Delta}({\bf D}^2 u) = \Delta^2 u$.

After applying integration by parts twice, we have

2.3.   Local VEM space $\mathcal{Q}_k(D)$

Then, for $k\geq 2,$ the local VEM space $\mathcal{Q}^k(D)\in H^2(D)$ is defined as: $v\in H^2(D)$ belongs to $\mathcal{Q}^k(D)$ if and only if (i) $v|_{\partial D}$ and trace of $\frac{\partial v}{\partial n}$ on $\partial D$ belongs to $\mathbb{P}_{r, s}(\partial D),$ then (ii) there exists a polynomial $q_v( = \Delta^2v) \in \mathbb{P}_k(D)$ such that

and (iii)

where $\Pi_{k, D}^0$ is the projection from $L^2(D)$ onto $\mathbb{P}_k(D)$.

Remark 1. It's obvious that $\mathbb{P}_k(D)$ is a subspace of $\mathcal{Q}^k(D).$ From (13), we have $\Pi_{k, D}^0v = \Pi_{k, D}^\Delta v,$ $k = 2, 3$.

The choice (ii) can be replaced by $q_v( = \Delta^2v) \in \mathbb{P}_{k-2}(D)$ in [11], also Lemma 2.9–Lemma 2.15, and Corollary 1–Corollary 2 are valid. The reason we chose $q_v\in \mathbb{P}_{k}(D)$ is that it helps to get the same error estimate with less smooth right hand side $f$. For $k = 2, 3$, we only require $f\in L^2(\Omega)$, while in [11], $f\in H^1(\Omega)$ for $k = 2$, and $f\in H^2(\Omega)$ for $k = 3$.

For completeness, we recall the definition for degrees of freedom in [10], employ the following notation: for $i$ a nonnegative integer, $e$ an edge with midpoint $x_e$, length $h_e$, the set of $i+1$ normalized monomials is denoted by $\mathcal{M}_i^e$

And for domain $D$ with diameter $h_D$ and barycenter ${\bf x}_D$, the set of $(i+1)(i+2)/2$ normalized monomials is defined by $\mathcal{M}_i^D$

where $\alpha$ is a nonnegative multiindex $\alpha = (\alpha_1, \alpha_2)$, $|\alpha| = \alpha_1+\alpha_2$ and ${\bf x}^\alpha = x_1^{\alpha_1} x_2^{\alpha_2}$. In $D$ the degrees of freedom are denoted as:

Lemma 2.4. Given any $g\in \mathbb{P}_{r, s}({\partial{D}})$ and $f\in \mathbb{P}_k(D)$, there exists a unique function $v\in H^2(D)$ such that (i) $v = g,$ $\frac{\partial v}{\partial n} = \frac{\partial g}{\partial n}$ on $\partial D$ and (ii)

Proof. Similar as in [7], let $\tilde{g} \in H^2(D)$ be a $C^1$, $\mathbb{P}_k$ piece-wise polynomial constructed in Section 2.6.1, such that $v = g,$ $\frac{\partial v}{\partial n} = \frac{\partial g}{\partial n}$ on $\partial D$. Then the unique $v\in H^2(D)$ is given by $\phi+\tilde{g}$, where $\phi\in H^2_0(D)$ is defined by

The proof is completed.

Lemma 2.5. We have (i) ${\text{dim}}$ $\mathcal{Q}^k(D)$ = ${\text{dim}}$ $\mathbb{P}_{r, s}(\partial D)$ $\rm+ dim$ $\mathbb{P}_{m}(D)$, and (ii) $v\in \mathcal{Q}^k(D)$ is uniquely determined by $v|_{\partial D}, \left.\frac{\partial v}{\partial n}\right|_{\partial D}$ and $\Pi_{k-4, D}^0v$.

Proof. Following [7] and [10], let

The linear map $v\mapsto (v|_{\partial D}, \left.\frac{\partial v}{\partial n}\right|_{\partial D}, \Delta^2 v)$ from ${\mathcal{Q}}^k_D$ to $\mathbb{P}_{r, s}(\partial D)\times \mathbb{P}_k(D)$ is an isomorphism by Lemma 2.4.

The linear map $v\mapsto (v|_{\partial D}, \left.\frac{\partial v}{\partial n}\right|_{\partial D}, \Pi_{k-4, D}^0 v +(\Pi_{k, D}^0 -\Pi_{k-4, D}^0)(v-\Pi_{k, D}^\Delta v) )$ from ${\mathcal Q^k}(D)$ to $\mathbb{P}_{r, s}(\partial D)\times \mathbb{P}_k(D)$ is also an isomorphism. Suppose $v\in$ null space, then $\Pi_{k-4, D}^0v = 0$,

With (10)-(8) and (11), we have $\Pi_{k, D}^\Delta v = 0$, so that by (13),

In (12), let $w = v\in \mathcal{Q}^k(D)$, then we have

This completes the proof.

Lemma 2.6. Discrete Estimates. From Conditions (1)-(2), and the equivalence of norms on finite dimensional vector spaces, for any $u\in \mathbb{P}_k$, we have the following estimates:

2.4.   Estimates of $|||\cdot|||_{k, D}$

The semi-norm $|||\cdot|||_{k, D}$ for $\xi\in H^2(D)$ is defined by

There is an obvious stability estimate from (10)

We define the kernel of operator $\Pi_{k, D}^\Delta$ as:

And for any $v \in\mathcal{Q}^k(D)$, we have

Lemma 2.7. For any $v \in \mathcal{Q}^k(D)$, we have the equivalence of norms:

Proof. Suppose $h_D = 1$, by the discrete estimates from Lemma 2.6, we have

and

so that the equivalence is obtained.

Lemma 2.8. For any $p\in \mathbb{P}_{k-4}$, $k\geq 2$, there exists $q \in \mathbb{P}_{k}$, such that $\Delta^2 q = p$ and

Proof. From [7], we know that $\Delta$ maps $\mathbb{P}_k$ to $\mathbb{P}_{k-2}$, so that $\Delta^2 = \Delta\Delta$ maps $\mathbb{P}_k$ to $\mathbb{P}_{k-4}$. Then there exists an operator $(\Delta^2)^{\dagger}: \mathbb{P}_{k-4}\rightarrow \mathbb{P}_{k}$ such that $\Delta^2 (\Delta^2)^{\dagger}$ is the identity operator on $\mathbb{P}_{k-4}$. We define the norm of $p$ as

Since we have

the minimization problem

is solvable. So, there exists $q = (\Delta^2)^{\dagger}p$ such that

By the equivalence of norms on finite dimensional vector space, we have

then the result is obtained.

Lemma 2.9. For any $v\in {\rm Ker} \Pi_{k, D}^\Delta$, we have

Proof. Suppose $h_D = 1$. For $k<4$, $\Pi_{k-4, D}^0 v = 0,$ the equivalence is trivial. For $k\geq 4,$ let $v\in {\rm Ker} \Pi_{k, D}^\Delta$, by (10), (11), and using the same $p$, $q$ in Lemma 2.8, we have

By lemma 2.6, we have

and $\left\|U_{nn}({{\bf D}^2 q})\right\|_{L^2(\partial D)} \lesssim \|{q}\|_{L^2(D)}$. Then, by Lemma 2.8,

so that

which means

with Lemma 2.7, we get the result.

Remark 2. Same as in [7], we have

where $\partial/\partial t$ denotes a tangential derivative along $\partial D$.

There is also a stability estimate for $\Pi_{k, D}^\Delta\xi$ in $H^1(D)$ norm in terms of the semi-norm $|||\cdot|||_{k, D}$.

2.5.   Estimates of $\Pi_{k, D}^\Delta$ and $\Pi_{k, D}^0$

Lemma 2.10. We have

Proof. Suppose $u = \Pi_{k, D}^\Delta\xi$ and $h_D = 1$, by (10), (11), we have

then, by Cauchy-Schwarz inequality, Lemma 2.6 and (19), we have

By (8)-(9), and Lemma 2.3, we have

also we have

Similarly,

From (23)-(27), the following inequality is valid

The proof is completed.

Lemma 2.11. For any $\xi\in H^2(D)$, we have

and there exists $\bar{\xi}\in \mathbb{P}_1$, such that

where

Proof. Assume $h_D$ = 1, then by trace theorem

So that we have (29), and by Lemma 2.3

with the definition of $\bar{\xi}$, we arrived at

The proof is completed.

Corollary 1. We have

Proof. From Lemma 2.1, Lemma 2.10, and Lemma 2.11, for any $q\in \mathbb{P}_l, \ \xi\in H^{l+1}(D)$ we have

so as the $H^1$ and $H^2$ error estimates.

For the $L^2$ operator $\Pi_{k, D}^0$, from [7], we have

and

Lemma 2.12. The estimates of $\Pi_{k, D}^0$ satisfy

Proof. Here, suppose $\xi\in H^2(D)$, then by (20), (32), (35) and the inverse inequality, we have

which completes the proof.

2.6.   An inverse inequality

Lemma 2.13. The following inequality is valid

for any $v\in \mathcal{Q}^k(D)$ and $\xi\in H^2(D)$ such that $\xi = v$ on $\partial D$, $\frac{\partial \xi}{\partial n} = \frac{\partial v}{\partial n}$ on $\partial D$, and $\Pi_{k, D}^0(\xi -v) = 0.$

Proof. In (11), let $u$ be $v$, $v$ be $\xi-v$, the second and third terms at right hand side can be eliminated. Also $\xi$ satisfies $\Pi_{k, D}^0(\xi-v) = 0$, and $U_{\Delta}({\bf D}^2v) \in \mathbb{P}_k(D)$ so that $((v, \xi-v))_D = 0$. Then by (10) and (11), we have

And hence,

which means

So the proof is completed. Next, we will consider the relation between $|v|_{H^2(D)}$ and $|||v|||_{k, D}$, $\forall v \in \mathcal{Q}^k(D)$.

2.6.1.   Construction of $w$

The degree of freedom of $v\in \mathcal{Q}^k(D)$ is defined in [10], from (4.7)–(4.11). For $k\geq 2,$ we will employ the triangulation $\mathcal{T}_D$ to define a piecewise polynomial $w$ which has the same boundary conditions as $v$. On each internal triangle, we employ a $\mathbb{P}_{r}$ macroelement, which is defined in [12], Section 1. Suppose $k = 2, 3$, in Figure 2, on each internal triangle, the function is defined by $\mathbb{P}_3$ macroelement as in Figure 3. All degrees of freedom within $D$ are 0.

From the definition of $w$, we have

Lemma 2.14.

Proof. Following [7], it suffices to prove when $h_D = 1$. Let $\phi$ be a smooth function supported on the disc $\mathfrak{B}$ with radius $\rho$, such that

By the equivalence of norms on finite dimensional spaces, we have

Let $w \in H^2 (D)$ be the piecewise polynomial constructed in Section 2.6.1, and let $\xi = w + p\phi$ for $p\in \mathbb{P}_{k}$ such that

or equivalently

Let $q = p$ in (40), then

and by Lemma 2.10,

From (39), we have

Also, by Lemma 2.13,

By (39) and (41),

Then, $\forall v \in \mathcal{Q}^k(D)$, we have

By Lemma 2.7, we arrived at the estimate.

Corollary 2. For any $v\in \mathcal{Q}^k(D)$,

Proof. From Lemma 2.10, Corollary 1 and Lemma 2.14, we have

The proof is completed.

2.7.   Estimate of interpolation operator

The interpolation operator $I_{k, D} : H^3 (D) \rightarrow \mathcal{Q}^k(D)$ is defined by the conditions that $\xi$ and $I_{k, D}\xi$ have the same value for each degree of freedom of $I_{k, D}\xi$. It is clear that

Lemma 2.15. The interpolation errors are listed below, $\forall \xi\in H^{l+1}(D), \ 1\leq l\leq k,$ we have

Proof. Suppose $h_D = 1$, by Trace theorem, Lemma 2.2, Lemma 2.10 and Corollary 2, we have

The results follow from Lemma 2.1, and $I_{k, D}q = q, \forall q\in \mathbb{P}_l$.

3.   The biharmonic problem in two dimensions

Let $\Omega$ be a bounded polygonal domain in $\mathbb{R}^2$, $f\in L^2(\Omega)$, the biharmonic equation is

The variational formulation of (45) is finding $u\in H^2_0(\Omega)$, such that

where

Remark 3. For $u, v\in H^2_0(\Omega)$, we also have $a(u, v) = \int_\Omega\Delta u\Delta v \ {\rm d}x$. However, $((u, v))_{D},$ where $D$ is a sub-domain of $\Omega$ and $\int_D\Delta u\Delta v \ {\rm d}x$ are not equivalent.

In following sections, we will use virtual element method to solve (45).

3.1.   Virtual element spaces

Let $\mathcal{T}_h$ be a conforming partition of $\Omega$ by polygonal subdomains, i.e., the intersection of two distinct subdomains is either empty, common vertices or common edges. We assume that all the polygons $D\in \mathcal{T}_h$ satisfy the shape regularity assumptions in Section 2.1.

We take the virtual element space $\mathcal{Q}_h^k$ to be $\{v \in H_0^2(\Omega): v|_D \in \mathcal{Q}^k(D), \ \forall D \in \mathcal{T}_h \}$ and denote by $\mathcal{P}_{h}^k$ the space of (discontinuous) piecewise polynomials of degree $\leq k$ with respect to $\mathcal{T}_h$. The operators are defined in terms of their local counterparts:

Also, the semi-norm on $H^2(\Omega)$ is defined as

so that $|v|_{2, h} = |v|_{H^2(D)}$ for $v\in H^2(D),$ and

The local estimates: Corollary 1, (34)-(35), Lemma 2.12 and Lemma 2.15 immediately imply the following global results, where $h = \max\limits_{D\in\mathcal{T}_h} h_D$.

Corollary 3. The global error estimates are listed below, $\forall \xi\in H^{l+1}(D), \ 1\leq l\leq k,$ we have

where the norm $\|\cdot\|: = \|\cdot\|_{L^2(\Omega)}$.

3.2.   The discrete problem

Our goal is to find $u_h\in \mathcal{Q}_h^k$, which satisfies

where $\Xi_h$ is an operator from $\mathcal{Q}_h^k$ to $\mathcal{P}_h^k$, and

for $w, v\in H^2(\Omega)$. If $w, v\in \mathcal{Q}^k(D),$ we have

so that $S^D(w, v)$ can be computed explicitly with the degrees of freedom of $\mathcal{Q}^k(D)$.

3.2.1.   Other choices of $S^D(\cdot, \cdot)$

The systems of virtual element method are equivalent if the bilinear form satisfies

From Lemma 2.9 and Remark 2, we can take

for $w, v\in {\rm Ker}\Pi_{k, D}^\Delta$.

3.3.   Well-posedness of the discrete problem

We can show the well-posedness by the following Lemmas.

Lemma 3.1. For any $v\in H^2(D)$, we have

Proof. By (10)-(8) and Lemma 2.11, let $w = v-\Pi_{k, D}^\Delta v$, we have $\bar{w} = 0$, and

Lemma 3.2. For any $v\in {\rm Ker}\Pi_{k, D}^\Delta$, we have $S^D(v, v)\approx |v|^2_{H^2(D)}.$

Proof. For any $v\in {\rm Ker}\Pi_{k, D}^\Delta,$ we have $v\in H^2(D)$ and

with Lemma 2.14 and Lemma 3.1, we have the equivalence.

Remark 4. By Lemma 3.1, for any $v, w\in \mathcal{Q}_h^k$, we have

Then by Lemma 3.2, as in [7], we have

which means problem (53) is uniquely solvable.

3.4.   Choice of $\Xi_h$

Here, we chose $\Xi_h$ as

The following result can be used for error analysis in $H^2(\Omega)$ norm. Define $H^0(\Omega) : = L^2(\Omega)$, and the indices $(k, l, m)$ as

Lemma 3.3. With (61), we have

Proof. For $k\leq 3, f\in L^2(\Omega)$, we define $\Pi_{k-4}^0 f = 0,$ so that with Corollary 3, we have

For $k\geq 4, f\in H^l(\Omega), l = k-3$, with Lemma 2.1 and Corollary 3, we have

Lemma 3.4. With (61), we have

for any $f\in H^{l}(\Omega), \ w\in H^3(\Omega), \ k\geq 2.$ And

for any $f\in H^{l}(\Omega), \ w\in H^{3+s}(\Omega), \ k\geq 3, \ 0<s\leq 1.$

Proof. Follow the proof in Lemma 3.3 with Lemma 2.12 and Lemma 2.15. For $k\geq 2$, $w\in H^3(\Omega)$, we have

then with Corollary 3, we can get (63), so as $k\geq 3$ and $w\in H^{3+s}(\Omega)$.

3.5.   Error estimate in $|\cdot|_{H^2(\Omega)}$ norm

Firstly, the error estimate in $|\cdot|_{H^2(\Omega)}$ norm for $u-u_h$ is given in Theorem 3.5.

Theorem 3.5. With (61), we have

Suppose $u\in H^{k+1}(\Omega)$ then

Proof. Similar as in [7], for any given $v_h\in \mathcal{Q}_h^k$, from (59), we have

and by (53),

Then from (54) to (56), by (33), (44), we have

with Lemma 3.3, Remark 4 or Lemma 3.1 and Corollary 3, the estimate is obtained.

3.6.   Error estimate in $|\cdot|_{H^1(\Omega)}$ and $\|\cdot\|_{L^2(\Omega)}$ norm

We suppose $\Omega$ is also convex and start with a consistency result.

Lemma 3.6. Suppose $u\in H^{k+1}(\Omega)$ and $l$ is defined in (61), then

for any $\xi\in H^3(\Omega)\cap H^2_0(\Omega), \ k\geq 2.$ And

for any $\xi\in H^{3+s}(\Omega)\cap H^2_0(\Omega), \ k\geq 3$.

Proof. Similar as in [7], we have

Then by Lemma 3.4, Corollary 3, and Theorem 3.5, we get the estimate.

From the regularity results of (45), see [13], we have

Theorem 3.7. Suppose $u\in H^{k+1}(\Omega)$ then

where $l$ is defined in (61).

Proof. Using the duality arguments and (65), let $f = -\Delta(u-u_h),$ and $\phi\in H_0^2(\Omega)$ be the solution of (45), then we get

by (67), we have $|\phi|_{H^3(\Omega)} \lesssim |u-u_h|_{H^1(\Omega)}$, then the result is obtained.

Theorem 3.8. Suppose $u\in H^{k+1}(\Omega)$ then

where $l$ is defined in (61).

Proof. For $k = 2,$ by Theorem 3.7 and Poincar$\acute{\rm e}$ inequality, (70) is obtained. For $k\geq 3$, using the duality arguments and (66), let $f = u-u_h,$ and $\phi\in H_0^2(\Omega)$ be the solution of (45), then we get

by (68), we have $\|\phi\|_{H^{3+s}(\Omega)} \lesssim \|u-u_h\|_{L^2(\Omega)}$, then the result is obtained.

3.7.   Error estimates for $\Pi_{k, h}^\Delta u_h$

We also have an error estimate for the computable $\Pi_{k, h}^\Delta u_h$.

Corollary 4. Suppose $u\in H^{k+1}(\Omega)$ then

Proof. By (20), Theorem 3.5, Corollary 3, and

the estimate is obtained.

Corollary 5. Suppose $u\in H^{k+1}(\Omega)$ then

where $l$ is defined in (61).

Proof. By Corollary 1, Corollary 3, Theorem 3.7, Lemma 2.10 and

Suppose $\xi = u-u_h,$ and $\bar{\xi}$ is defined as in Lemma 2.11, the second term is estimated as

so that

sum them up then the estimate is obtained.

Corollary 6. Suppose $u\in H^{k+1}(\Omega)$ then

where $l$ is defined in (61).

Proof. By Corollary 1, Corollary 3, Theorem 3.7, Theorem 3.8, Lemma 2.10 and

From Lemma 2.11, the second term is estimated as

sum them up then, the estimate is obtained.

3.8.   Error estimates for $\Pi_{k, h}^0u_h$

Since $\Pi_{k, h}^0u_h$ can be computed explicitly, we can also get the similar error estimates between $u$ and $\Pi_{k, h}^0u_h$.

Corollary 7. Suppose $u\in H^{k+1}(\Omega)$ then

Proof. By Lemma 2.12, Theorem 3.5, and

the estimate is obtained.

Corollary 8. Suppose $u\in H^{k+1}(\Omega)$ then

where $l$ is defined in (61).

Proof. By (34), (35), Theorem 3.7, and

for the second term, we have

sum them up then the estimate is obtained.

Corollary 9. Suppose $u\in H^{k+1}(\Omega)$ then

where $l$ is defined in (61).

Proof. By Theorem 3.8, and

the second term is estimated as

so the results are obtained.

4.   Conclusion

We have extended the works done in [7] to forth order problems in two dimension. Similar basic estimates for local projections $\Pi_{k, D}^\Delta$, $\Pi_{k, D}^0$, $I_{k, D}$ and the improved error analysis of modified virtual element method for biharmonic equation are obtained. The computable piecewise polynomials $\Pi_{k, h}^\Delta u_h$ and $\Pi_{k, h}^0 u_h$ are more efficient to use in practice.

We can replace (9) by (76)

To compute (76), we have

For $k\geq 4$, can replace (8) by (78)

And the replacements attain same estimates for projections and error analysis.

Acknowledgments

We would like to thank anonymous reviewers for their valuable comments and suggestions. Also we are grateful for the editors' patience while handling this paper.