Obtaining higher-order Galerkin accuracy when the boundary is polygonally approximated

1. Introduction

When a Dirichlet problem on a smooth domain is approximated by a polygon, an error occurs that is suboptimal for quadratic approximation [1, 10, 11]. However, this can be corrected by a modification of the variational form [2]. Here we review this approach and suggest a new one.

Let $Ω$ be a smooth, bounded, two-dimensional domain. Consider the Poisson equation with Dirichlet boundary conditions:

(1)

- Δ u = f in Ω, u = g on \partial Ω .

We assume that $f$ and $g$ are sufficiently smooth that $u$ can be extended to be in $H^{k + 1} (ˆ Ω)$ , where $ˆ Ω$ contains a neighborhood of the closure of $Ω$ .

One way to discretize (1) is to approximate the domain $Ω$ by polygons $Ω_{h}$ , where the edge lengths of $\partial Ω_{h}$ are of order $h$ in size. Then conventional finite elements can be employed, with the Dirichlet boundary conditions being approximated by the assumption that $u_{h} =^g$ on $\partial Ω_{h}$ [3], with $^g$

appropriately defined. For example, let us suppose for the moment that

g \equiv 0

and we take

^g \equiv 0

as well. In particular, we assume that

Ω_{h}

is triangulated with a quasi-uniform mesh

T_{h}

of maximum triangle size

h

, and the boundary vertices of

Ω_{h}

are in

\partial Ω

. We define

{˚ W}_{h}^{k} := H_{0}^{1} (Ω) \cap W_{h}^{k}

where

W_{h}^{k} = {v \in C (Ω_{h}) : v |_{T} \in P_{k} (T), \forall T \in T_{h}} .

Then the standard finite element approximation finds $u_{h} \in {˚ W}_{h}^{k}$ satisfying

(2)

a_{h} (u_{h}, v) = (f, v)_{L^{2} (Ω_{h})}, \forall v \in {˚ W}_{h}^{k},

where $a_{h} (u, v) := \int_{Ω_{h}} \nabla u \cdot \nabla v d x$ . Here we assume that $f$ is extended smoothly outside of $Ω$ .

This approach for $k = 1$

(piecewise linear approximation) leads to the error estimate

∥ u - u_{h} ∥_{H^{1} (Ω_{h})} \leq C h ∥ u ∥_{H^{2} (^Ω)} .

However, when this approach is applied with piecewise quadratic polynomials ( $k = 2$ ), the best possible error estimate is

(3)

∥ u - u_{h} ∥_{H^{1} (Ω_{h})} \leq C h^{3 / 2},

which is less than optimal order by a factor of $\sqrt{h}$ . The reason of course is that we have made only a piecewise linear approximation of $\partial Ω$ . Table 1 summarizes some computational experiments for the test problem in Section 2.1. We see a significant improvement for quadratics over linears, but there is almost no improvement with cubics. Moreover, we will see that a significant improvement using quadratics can be obtained using simple approaches that modify the variational form.

There have been many techniques introduced to circumvent the loss of accuracy with quadratics (and higher-order piecewise polynomials) [11, 6]. However, all of them require some modification of the quadrature for the elements at the boundary.

Here we review an approach by Bramble et al. [2] that solves directly on $Ω_{h}$ , but with a modified variational form based on the method of Nitsche [6]. The method [2] has been modified and applied in many ways [4]. However, the method in [2] leads to a non symmetric bilinear form. Given this shortcoming we define a new method that is symmetric and solves the problem on $Ω_{h}$ that has similar convergence results. As we will see in the next section, one main idea in [2] is that one uses a Taylor series of the solution near the boundary to define appropriate boundary conditions on $\partial Ω_{h}$ . We should mention that this idea has been used recently (see for example [5, 8]).

$k$	$M$	L2 err	rate	H1 err	rate	seg	hmax
1	2	1.84e+00	NA	6.25e+00	NA	10	1.05e+00
1	4	2.93e-01	2.65	1.89e+00	1.73	20	4.94e-01
1	8	9.55e-02	1.62	1.06e+00	0.83	40	2.61e-01
1	16	2.47e-02	1.95	5.45e-01	0.96	80	1.35e-01
2	2	4.18e-01	NA	1.41e+00	NA	10	1.05e+00
2	4	9.44e-02	2.15	4.26e-01	1.73	20	4.94e-01
2	8	2.30e-02	2.04	1.59e-01	1.42	40	2.61e-01
2	16	5.62e-03	2.03	5.45e-02	1.54	80	1.35e-01
3	2	3.17e-01	NA	8.25e-01	NA	10	1.05e+00
3	4	8.81e-02	1.85	2.94e-01	1.49	20	4.94e-01
3	8	2.22e-02	1.99	1.07e-01	1.46	40	2.61e-01
3	16	5.53e-03	2.01	3.82e-02	1.49	80	1.35e-01

Table 1. Errors

u_{h} - u_{I}

L^{2} (Ω_{h})

and

H^{1} (Ω_{h})

, as a function of the maximum mesh size (hmax) for the polygonal approximation (2) for test problem in Section 2.1 using various polynomial degrees

k

. Key: “

M

” is input parameter to mshr function circle used to generate the mesh, “seg” is the number of boundary edges. The approximate solutions were generated using (2).

2. The Bramble-Dupont-Thomée approach

Figure 1. Definitions of (a) $δ$ and (b) $d$ .

The method [2] of Bramble-Dupont-Thomée (BDT) achieves high-order accuracy by modifying Nitsche’s method [6] applied on $Ω_{h}$ . We assume that $Ω_{h} \subset Ω$ and we do not necessarily assume that the boundary vertices of $Ω_{h}$ belong to $\partial Ω$ . The bilinear form used in [2] is

(4)

N_{h} (u, v) = a_{h} (u, v) - \int_{\partial Ω_{h}} \frac{\partial u}{\partial n} v d s - \int_{\partial Ω_{h}} (u + δ \frac{\partial u}{\partial n}) (\frac{\partial v}{\partial n} - γ h^{- 1} v) d s

Here, $n$ denotes the outward-directed normal to $\partial Ω_{h}$ and

δ (x) = min {s > 0 : x + s n \in \partial Ω} .

Contrast the definition of $δ$ to the closely related function $d$ defined by

d (x) = min {| x - y | : y \in \partial Ω} .

For simplicity the assume that $g = 0$ . Then the BDT method will find $u_{h} \in W_{h}^{k}$ such that

N_{h} (u_{h}, v) = \int_{Ω_{h}} f v d x for all v \in W_{h}^{k} .

If $δ$ were 0, this would be Nitsche’s method on $Ω_{h}$ .

Corrections of arbitrary order, involving terms $δ^{ℓ} \frac{\partial^{ℓ} u}{\partial n^{ℓ}}$ for $ℓ > 1$ are studied in [2], but for simplicity we restrict attention to the first-order correction to Nitsche’s method given in (4). The error estimates obtained in [2] are as follows

| | | u - u_{h} | | |_{1} \leq C h^{k} ∥ u ∥_{H^{k + 1} (Ω)} + C h^{7 / 2} ∥ u ∥_{W_{\infty}^{2} (Ω)},

where

| | | v | | |_{1} := (a_{h} (v, v) + h^{- 1} \int_{\partial Ω_{h}} v^{2} d s + h \int_{\partial Ω_{h}} (\frac{\partial v}{\partial n})^{2} d s)^{1 / 2} .

Thus using the variational form (4) leads to an approximation that is optimal-order with quadratics and cubics and is only suboptimal for quartics by a factor of $\sqrt{h}$ .

Figure 2. Errors $u_{h} - u_{I}$ in (a) $L^{2} (Ω_{h})$ and (b) $H^{1} (Ω_{h})$ as a function of the maximum mesh size for the BDT method with $γ = 100$ . The asterisks indicate data for (a) $k = 4$ and (b) $k = 5$ .

2.1. An example of a circle

We consider a numerical example. Consider the case where $Ω$ is a disc of radius $R$ centered at the origin, in which case we have $d (x) = R - | x |$ . However, it is more difficult to evaluate $δ (x)$ . We have $x + δ (x) n \in \partial Ω$ for $x \in \partial Ω_{h}$ , where $n$ denotes the outward normal to $Ω_{h}$ . We can write $x = (x \cdot n) n + (x \cdot t) t$ , and $(x \cdot t)^{2} = | x |^{2} - (x \cdot n)^{2}$ . Since $| x + δ (x) n | = R$ , we have

R^{2} = (x \cdot t)^{2} + (x \cdot n + δ (x))^{2} = | x |^{2} - (x \cdot n)^{2} + ((x \cdot n + δ (x))^{2} .

Then

δ (x) = \pm \sqrt{R^{2} - | x |^{2} + (x \cdot n)^{2}} - x \cdot n .

Note that for $x \in \partial Ω_{h}$ , $| x | \leq R$ and $x \cdot n > 0$ . Since $δ (x) \geq 0$ , we must pick the plus sign, so

δ (x) = \sqrt{R^{2} - | x |^{2} + (x \cdot n)^{2}} - x \cdot n .

It is not hard to see that $d - δ = O (h^{4})$ in this case.

This problem is simple to implement using the FEniCS Project code dolfin [7]. We take $R = 1$ , $u (x, y) = 1 - (x^{2} + y^{2})^{3}$ , and $f = 36 (x^{2} + y^{2})^{2}$ in the computational experiments described subsequently. Computational results for this example are given in Table 2 where we see optimal order approximation for $k \leq 3$ , improvement for $k = 4$ over $k = 3$ (suboptimal by a factor $h^{- 1 / 2}$ ), and no improvement for quintics. These errors are depicted in Figure 2.

$k$	$M$	hmax	L2 error	rate	H1 error	rate
1	8	0.261	0.0947	1.61	1.06	0.82
1	16	0.135	0.0245	1.95	0.544	0.96
1	32	0.0688	0.00639	1.94	0.277	0.97
1	64	0.0353	0.00158	2.02	0.137	1.02
2	8	0.261	2.81e-03	2.61	0.103	1.57
2	16	0.135	3.70e-04	2.93	0.0277	1.89
2	32	0.0688	4.77e-05	2.96	0.00717	1.95
2	64	0.0353	5.91e-06	3.01	0.00179	2.00
3	8	0.261	1.56e-04	3.92	5.31e-03	2.54
3	16	0.135	9.44e-06	4.05	7.06e-04	2.91
3	32	0.0688	5.81e-07	4.02	9.23e-05	2.94
3	64	0.0353	3.57e-08	4.02	1.15e-05	3.00
4	8	0.261	1.49e-04	3.96	7.41e-04	3.42
4	16	0.135	9.29e-06	4.00	6.63e-05	3.48
4	32	0.0688	5.80e-07	4.00	5.90e-06	3.49
4	64	0.0353	3.63e-08	4.00	5.22e-07	3.50
5	8	0.261	1.47e-04	3.96	7.10e-04	3.41
5	16	0.135	9.27e-06	3.99	6.44e-05	3.46
5	32	0.0688	5.80e-07	4.00	5.77e-06	3.48
5	64	0.0353	3.62e-08	4.00	5.12e-07	3.49

Table 2. Errors

u_{h} - u_{I}

L^{2} (Ω_{h})

and

H^{1} (Ω_{h})

as a function of mesh size (hmax) for the the BDT approximation in Section 2, with

γ = 100

, for various polynomial degrees

k

. Key:

M

is the value of the meshsize input parameter to the mshr function circle used to generate the mesh. The number of boundary edges was set to

5 M

, and hmax is the maximum mesh size.

3. A new method based on a Robin-type approach

One issue with the BDT method is that the resulting linear system is not symmetric, although it is possible to symmetrize the method as we discuss in Section 8. Here we develop a technique that leads to a symmetric system. Moreover, this method does not require the parameter(s) from Nitsche’s method. For Nitsche’s method to succeed, $γ$ must be chosen appropriately [9].

We first separate $\partial Ω$ to its piecewise linear part and its curvilinear part. We will assume that $\partial Ω = Γ^{0} \cup S_{1} \cup \dots S_{ℓ}$ where $Γ^{0}$ is a piecewise linear segment and $S_{i}^{'} s$ are $C^{2}$ and no where linear. We let the end points of $S_{i}$ to be $y_{i - 1}, y_{i}$ .

For the method in this section we assume that the vertices of $Ω_{h}$ belong to $\partial Ω$ and hence $Ω_{h}$ might not be a subdomain of $Ω$ . Hence, we need to define $δ$ in this case. We assume that for every $x \in \partial Ω_{h}$ that is there is a unique smallest number $δ (x)$ in absolute value such that $Ω ∖ Γ^{0}$

x + δ (x) n (x) \in \partial Ω .

We assume that the approximate domain boundary $\partial Ω_{h}$ can be decomposed into three parts, as follows. Let $E_{h}$ be the edges of $\partial Ω_{h}$ .

(5)

Γ^{\pm} = \cup {e \in E_{h} : \pm δ |_{e^{o}} > 0},

where $e^{o}$ denotes the interior of $e$ . Let $Γ = Γ^{+} \cup Γ^{-}$ . We assume the following.

Assumption 1.

We assume that all the vertices of $\partial Ω_{h}$ belong to $\partial Ω$ and that each $y_{i}$ (for $0 \leq i \leq ℓ$ ) is a vertex of $\partial Ω_{h}$ . Finally, we assume that

\partial Ω_{h} = Γ^{0} \cup Γ .

Our method is based on a Robin type of boundary condition on $Γ$ . In fact, our method will be based on the closely related problem:

$- Δ w =$	$f,$	$on Ω,$
$w =$	$0,$	$on Γ^{0},$
$w + δ \frac{\partial w}{\partial n} =$	$^g,$	$on Γ .$

Here we define $^g (x) = g (x + δ (x) n (x))$ for $x \in Γ$ and not a vertex of $\partial Ω_{h}$ . The key here is that, using that $u$ vanishes on $\partial Ω$ , for $x \in Γ$ ( $x$ not a vertex of $\partial Ω_{h}$ ) we have

(6)

u (x) + δ \frac{\partial u}{\partial n} (x) =^g (x) - \frac{δ^{2}}{2} \partial_{n n} u (z),

where $z$ lies in the line segment with end points $x$ and $x + δ (x) n (x)$ .

Now we can write the method. We start by defining the finite element space we will use

V_{h}^{k} = {W_{h}^{k} : v = 0 on Γ^{0}, v (x) = 0 for all vertices % of x of \partial Ω_{h}} .

Also define

V_{h}^{k} (g) = {W_{h}^{k} : v = g_{I} on Γ^{0}, v (x) = I g (x) for all% vertices of x of \partial Ω_{h}} .

where $g_{I} \in C (\partial Ω_{h})$ is a suitable approximation of $g$ and is a piecewise polynomial of degree at most $k$ on $\partial Ω_{h}$ .

The bilinear form is given by

b_{h} (u, v) := a_{h} (u, v) + c_{h} (u, v),

where

c_{h} (u, v) = \int_{Γ} δ^{- 1} u v d s .

Then the method solves:

Find $u_{h} \in V_{h}^{k} (g)$ such that

(7)

b_{h} (u_{h}, v) = \int_{Ω_{h}} F v + \int_{Γ} δ^{- 1}^g v d s . for all v \in V_{h}^{k} .

Here

F = {\begin{matrix} f & on Ω \cap Ω_{h} I^{1} f & on Ω_{h} ∖ Ω, \end{matrix}

where $I^{1}$

is the linear interpolant onto

W_{h}^{1}

. Note that we can define

I^{1} f

only knowing

f

Ω

. Alternatively, if we have an analytic representation of

f

we can define

F

as a smooth extension of

f

outside of

Ω

4. Error Analysis

4.1. Stability Analysis

Unfortunately, the bilinear form $b_{h}$ is not positive definite. However, we will be able to prove stability of method. In order to do so, we need to decompose the space $V_{h}^{k}$ into its boundary contribution and interior contribution. More precisely, we can write

V_{h}^{k} = {˚ W}_{h}^{k} \oplus B_{h}^{k},

where $B_{h}^{k} = {v \in V_{h}^{k} : v (x) = 0 for all interior Lagrange % points x}$ . We will define a norm on $V_{h}^{k}$ :

∥ v ∥_{a}^{2} := a_{h} (v, v)

and a semi-norm

| v |_{c}^{2} := \int_{Γ} \frac{v^{2}}{| δ |} d s .

Note that $| \cdot |_{c}$ is in fact a norm on $B_{h}^{k}$ .

The following crucial lemma will allow us to prove stability.

Lemma 1.

There exists a constant $c_{1} > 0$ such that

(8)

∥ v ∥_{a} \leq c_{1} \sqrt{h} | v |_{c} for all v \in B_{h}^{k} .

Proof.

Let $E_{h}^{Γ}$ be the collection of edges that are a subset of $Γ$ and let $T_{h}^{Γ}$ be triangles $T$ such that $T$ has an edge in $E_{h}^{Γ}$ . Then, if $v \in B_{h}^{k}$ and using inverse estimates we have

∥ v ∥_{a}^{2} = \sum T \in T_{h}^{Γ} ∥ \nabla v ∥_{L^{2} (T)}^{2} \leq \sum T \in T_{h}^{Γ} \frac{C}{h_{T}^{2}} ∥ v ∥_{L^{2} (T)}^{2} \leq \sum e \in E_{h}^{Γ} \frac{C}{h_{e}} ∥ v ∥_{L^{2} (e)}^{2} .

The result is complete after we use that ${max}_{x \in e} | δ (x) | \leq C h_{e}^{2}$ for $e \in E_{h}^{Γ}$ . ∎

We note that $c_{h} (u, v)$ may not be well defined for all $u, v \in V_{h}^{k}$ . Therefore, we need to make an assumption on $δ$ such that this is not the case.

Assumption 2.

We assume that $δ$ is such that

(9)

| c_{h} (u, v) | < \infty \forall u, v \in V_{h}^{k} .

For example, if $δ$ has a lower bound as follows, then (9) will hold. Suppose that the end points of $e \in E_{h}^{Γ}$ are $x_{0}$ and $x_{1}$ . Then we assume that there exists a constant $c > 0$ and a $p < 3$ such that

| x - x_{0} |^{p} | x - x_{1} |^{p} \leq c | δ (x) | for all x \in e,

where $c$ is independent of $e \in E_{h}^{Γ}$ . Under these conditions, Assumption 2 holds.

We can now prove the stability result.

Theorem 1.

We assume that Assumption 1 and Assumption 2 hold. Suppose that $G$ is a bounded linear function on $V_{h}^{k}$ and suppose that $u_{h} \in V_{h}^{k}$ solves

b_{h} (u_{h}, v) = G (v), for all v \in V_{h}^{k} .

Then, assuming $c_{1} \sqrt{h} \leq \frac{1}{2}$ we have

∥ u_{h} ∥_{a} \leq 2 ⎛ ⎜ ⎝ sup v_{h} \in {˚ W}_{h}^{k} \frac{| G (v_{h}) |}{∥ v_{h} ∥_{a}} ⎞ ⎟ ⎠ + \frac{11}{3} c_{1} \sqrt{h} ⎛ ⎝ sup v_{h} \in B_{h}^{k} \frac{| G (v_{h}) |}{| v_{h} |_{c}} ⎞ ⎠ .

and

| u_{h} |_{c} \leq \frac{3}{2} ⎛ ⎜ ⎝ sup v_{h} \in {˚ W}_{h}^{k} \frac{| G (v_{h}) |}{∥ v_{h} ∥_{a}} ⎞ ⎟ ⎠ + \frac{5}{3} ⎛ ⎝ sup v_{h} \in B_{h}^{k} \frac{| G (v_{h}) |}{| v_{h} |_{c}} ⎞ ⎠ .

Proof.

We know we can write $u_{h} = w_{h} + s_{h}$ where $w_{h} \in {˚ W}_{h}^{k}$ and $s_{h} \in B_{h}^{k}$ . Define $ϕ_{h} \in B_{h}^{k}$ by

ϕ_{h} = ⎧ ⎨ ⎩ \begin{matrix} s_{h} & on Γ^{+} - s_{h} & on Γ^{-} 0 & on Γ^{0} . \end{matrix}

Note that $| ϕ_{h} |_{c} = | s_{h} |_{c}$ . Now we can estimate $s_{h}$ .

| s_{h} |_{c}^{2} = c_{h} (s_{h}, ϕ_{h}) = b_{h} (u_{h}, ϕ_{h}) - a_{h} (u_{h}, ϕ_{h}) = G (ϕ_{h}) - a_{h} (u_{h}, ϕ_{h}) .

Hence, we have

	$\| s_{h} \|_{c}^{2} \leq$	$⎛ ⎝ sup v_{h} \in B_{h}^{k} \frac{\| G (v_{h}) \|}{\| v_{h} \|_{c}} ⎞ ⎠ \| ϕ_{h} \|_{c} + ∥ u_{h} ∥_{a} ∥ ϕ_{h} ∥_{a}$
	$\leq$	$⎛ ⎝ sup v_{h} \in B_{h}^{k} \frac{\| G (v_{h}) \|}{\| v_{h} \|_{c}} ⎞ ⎠ \| s_{h} \|_{c} + c_{1} \sqrt{h} (∥ w_{h} ∥_{a} + c_{1} \sqrt{h} \| s_{h} \|_{c}) \| s_{h} \|_{c} .$

Here we used (8) twice. In particular, we used $∥ u_{h} ∥_{a} \leq ∥ w_{h} ∥_{a} + ∥ s_{h} ∥_{a} \leq ∥ w_{h} ∥_{a} + c_{1} \sqrt{h} | s_{h} |_{c}$ . Assuming $h c_{1}^{2} \leq \frac{1}{4}$ we have

\frac{3}{4} | s_{h} |_{c}^{2} \leq ⎛ ⎝ sup v_{h} \in B_{h}^{k} \frac{| G (v_{h}) |}{| v_{h} |_{c}} ⎞ ⎠ | s_{h} |_{c} + \frac{1}{2} ∥ w_{h} ∥_{a} | s_{h} |_{c} .

Hence,

(10)

| s_{h} |_{c} \leq \frac{4}{3} ⎛ ⎝ sup v_{h} \in B_{h}^{k} \frac{| G (v_{h}) |}{| v_{h} |_{c}} ⎞ ⎠ + \frac{2}{3} ∥ w_{h} ∥_{a}

Next,

∥ w_{h} ∥_{a}^{2} = a_{h} (w_{h}, w_{h}) = a_{h} (u_{h}, w_{h}) - a_{h} (s_{h}, w_{h}) = b_{h} (u_{h}, w_{h}) - a_{h} (s_{h}, w_{h}) = G (w_{h}) - a_{h} (s_{h}, w_{h}) .

We therefore have

∥ w_{h} ∥_{a}^{2} \leq ⎛ ⎜ ⎝ sup v_{h} \in {˚ W}_{h}^{k} \frac{| G (v_{h}) |}{| v_{h} |_{a}} ⎞ ⎟ ⎠ ∥ w_{h} ∥_{a} + ∥ s_{h} ∥_{a} ∥ w_{h} ∥_{a} .

Hence, we obtain using (10)

	$∥ w_{h} ∥_{a} \leq$	$⎛ ⎜ ⎝ sup v_{h} \in {˚ W}_{h}^{k} \frac{\| G (v_{h}) \|}{∥ v_{h} ∥_{a}} ⎞ ⎟ ⎠ + ∥ s_{h} ∥_{a}$
		$\leq ⎛ ⎜ ⎝ sup v_{h} \in {˚ W}_{h}^{k} \frac{\| G (v_{h}) \|}{∥ v_{h} ∥_{a}} ⎞ ⎟ ⎠ + c_{1} \sqrt{h} ∥ s_{h} ∥_{c}$
		$\leq ⎛ ⎜ ⎝ sup v_{h} \in {˚ W}_{h}^{k} \frac{\| G (v_{h}) \|}{∥ v_{h} ∥_{a}} ⎞ ⎟ ⎠ + \frac{4}{3} c_{1} \sqrt{h} ⎛ ⎝ sup v_{h} \in B_{h}^{k} \frac{\| G (v_{h}) \|}{\| v_{h} \|_{c}} ⎞ ⎠ + \frac{1}{3} ∥ w_{h} ∥_{a}$

Thus we arrive at

∥ w_{h} ∥_{a} \leq \frac{3}{2} ⎛ ⎜ ⎝ sup v_{h} \in {˚ W}_{h}^{k} \frac{| G (v_{h}) |}{∥ v_{h} ∥_{a}} ⎞ ⎟ ⎠ + 2 c_{1} \sqrt{h} ⎛ ⎝ sup v_{h} \in B_{h}^{k} \frac{| G (v_{h}) |}{| v_{h} |_{c}} ⎞ ⎠ .

From this and (10) we get

| u_{h} |_{c} = | s_{h} |_{c} \leq \frac{3}{2} ⎛ ⎜ ⎝ sup v_{h} \in {˚ W}_{h}^{k} \frac{| G (v_{h}) |}{∥ v_{h} ∥_{a}} ⎞ ⎟ ⎠ + \frac{5}{3} ⎛ ⎝ sup v_{h} \in B_{h}^{k} \frac{| G (v_{h}) |}{| v_{h} |_{c}} ⎞ ⎠ .

Finally,

	$∥ u_{h} ∥_{a} \leq$	$∥ w_{h} ∥_{a} + ∥ s_{h} ∥_{a} \leq ∥ w_{h} ∥_{a} + c_{1} \sqrt{h} ∥ s_{h} ∥_{c}$
	$\leq$	$2 ⎛ ⎜ ⎝ sup v_{h} \in {˚ W}_{h}^{k} \frac{\| G (v_{h}) \|}{∥ v_{h} ∥_{a}} ⎞ ⎟ ⎠ + \frac{11}{3} c_{1} \sqrt{h} ⎛ ⎝ sup v_{h} \in B_{h}^{k} \frac{\| G (v_{h}) \|}{\| v_{h} \|_{c}} ⎞ ⎠ .$

∎

We can now prove error estimates after we make an assumption more stringent than Assumption 2.

Assumption 3.

Suppose that the end points of $e \in E_{h}^{Γ}$ are $x_{0}$ and $x_{1}$ . Then we assume that there exists a constant $β > 0$ such that

| x - x_{0} | | x - x_{1} | \leq β | δ (x) | for all x \in e,

where $β$ is independent of $e \in E_{h}^{Γ}$ .

Note that this assumption does not allow $\partial Ω$ and $\partial Ω_{h}$ to be tangent on the vertices of $Γ$ . Assumption 3 implies Assumption 2; in particular, the example after Assumption 2 holds with $p = 1$ .

Theorem 2.

We assume Assumptions 1 and 3 hold. We assume that $u$ solves (1) and belongs to $u \in W^{s, \infty} (Ω)$ where $s = max {k + 1, 4}$ . We assume that $g_{I} = u_{I} |_{\partial Ω_{h}}$ where $u_{I} \in W_{h}^{k}$ is the Lagrange interpolant of $u$ . Let $u_{h} \in V_{h}^{k} (g)$ solve (7) and assume that $u$ solves (1) then we have

	$∥ u - u_{h} ∥_{a} \leq$	$C h^{k} ∥ u ∥_{H^{k + 1} (^Ω)} + C h^{k + 1 / 2} ∥ u ∥_{W^{k + 1, \infty} (Γ)}$
		$+ C (h^{4} ∥ u ∥_{W^{4, \infty} (^Ω)} + h^{7 / 2} ∥ u ∥_{W^{2, \infty} (^Ω)}) .$

and

	$\| u - u_{h} \|_{c} \leq$	$C h^{k} ∥ u ∥_{H^{k + 1} (^Ω)} + C h^{k} ∥ u ∥_{W^{k + 1, \infty} (Γ)}$
		$+ C (h^{4} ∥ u ∥_{W^{4, \infty} (^Ω)} + h^{3} ∥ u ∥_{W^{2, \infty} (^Ω)}) .$

Proof.

We let $e_{h} = u_{I} - u_{h} \in V_{h}^{k}$ . Then we see that

b_{h} (e_{h}, v) = G (v) for all v \in V_{h}^{k},

where $G (v) = G_{1} (v) + G_{2} (v)$ , $G_{1} (v) = \int_{Ω_{h}} F v d x - b_{h} (u, v)$ and $G_{2} (v) = b_{h} (u - u_{I}, v)$ .

Note that using integration by parts we have

	$G_{1} (v) =$	$\int_{Ω_{h}} F v + \int_{Γ} δ^{- 1}^g v d s - \int_{Ω_{h}} (- Δ u) v d x - \int_{Γ} (\frac{\partial u}{\partial n} + \frac{1}{δ} (u -^g)) v$
	$=$	$\int_{Ω_{h} ∖ Ω} (I^{1} (- Δ u) - (- Δ u)) v d x - \int_{Γ} (\frac{\partial u}{\partial n} + \frac{1}{δ} (u -^g)) v .$

First consider $v \in {˚ W}_{h}^{k}$ then we have

| G_{1} (v) | \leq h^{2} ∥ u ∥_{W^{4, \infty} (^Ω)} ∥ v ∥_{L^{1} (Ω_{h} ∖ Ω)}

However, we have

	$∥ v ∥_{L^{1} (Ω_{h} ∖ Ω)} \leq$	$C h^{2} ∥ v ∥_{L^{\infty} (Ω_{h} ∖ Ω)}$
	$\leq$	$C h^{3} ∥ \nabla v ∥_{L^{\infty} (Ω)}$
	$\leq$	$C h^{2} ∥ \nabla v ∥_{L^{2} (Ω)} = C h^{2} ∥ v ∥_{a} .$

Therefore, we get

sup v \in {˚ W}_{h}^{k} \frac{| G_{1} (v) |}{| v |_{a}} \leq C h^{4} ∥ u ∥_{W^{4, \infty} (^Ω)} .

∎

Now consider $v \in B_{h}^{k}$ .

	$G_{1} (v) =$	$h^{4} ∥ u ∥_{W^{4, \infty} (^Ω)} ∥ v ∥_{L^{\infty} (Γ)} + ∥ δ^{- 1 / 2} (δ \frac{\partial u}{\partial n} + u -^g) ∥_{L^{\infty} (Γ)} ∥ v ∥_{c}$
	$\leq$	$h^{7 / 2} ∥ u ∥_{W^{4, \infty} (^Ω)} ∥ v ∥_{L^{2} (Γ)} + h^{3} ∥ u ∥_{W^{2, \infty} (^Ω)} ∥ v ∥_{c}$
	$\leq$	$h^{9 / 2} ∥ u ∥_{W^{4, \infty} (^Ω)} \| v \|_{c} + h^{3} ∥ u ∥_{W^{2, \infty} (^Ω)} ∥ v ∥_{c} .$

Here we used (6).

Hence,

\sqrt{h} (sup v \in B_{h}^{k} \frac{| G_{1} (v) |}{| v |_{c}}) \leq C (h^{7 / 2} ∥ u ∥_{W^{2, \infty} (^Ω)} + h^{5} ∥ u ∥_{W^{4, \infty} (^Ω)}) .

Now lets consider $G_{2}$ . If we let $v \in {˚ W}_{h}^{k}$ then

G_{2} (v) = a_{h} (u - u_{I}, v) \leq ∥ u - u_{I} ∥_{a} ∥ v ∥_{a}

Hence,

sup v \in {˚ W}_{h}^{k} \frac{| G_{2} (v) |}{∥ v ∥_{a}} \leq C h^{k} ∥ u ∥_{H^{k} (^Ω)} .

Now let $v \in B_{h}^{k}$ we then have

G_{2} (v) = ∥ u - u_{I} ∥_{a} ∥ v ∥_{a} + | u - u_{I} |_{c} | v |_{c} \leq c_{1} \sqrt{h} ∥ u - u_{I} ∥_{a} ∥ v ∥_{c} + | u - u_{I} |_{c} | v |_{c} .

Let $e \in E_{h}$ , $e \subset Γ$ with end points $x_{0}$ and $x_{1}$ . Then, we have $| (u - u_{I}) (x) |^{2} \leq C | x - x_{0} | | x - x_{1} | ∥ \partial_{t} (u - u_{I}) ∥_{L^{\infty} (e)}$ . Hence, using Assumption 3 we get

\frac{(u - u_{I})^{2} (x)}{| δ (x) |} \leq C β ∥ \partial_{t} (u - u_{I}) ∥_{L^{\infty} (e)} .

Thus,

\int_{e} \frac{(u - u_{I})^{2}}{| δ |} d s \leq C β | e | ∥ \partial_{t} (u - u_{I}) ∥_{L^{\infty} (e)}^{2} .

We then obtain the following estimate, after summing over all edges $e \subset Γ$ ,

| u - u_{I} |_{c}^{2} \leq C ∥ \partial_{t} (u - u_{I}) ∥_{L^{\infty} (Γ)}^{2} .

We get the following inequality after using approximation properties of the Lagrange interpolant:

| u - u_{I} |_{c} \leq C h^{k} ∥ u ∥_{W^{k + 1, \infty} (Γ)} .

Therefore, we have

\sqrt{h} sup v \in B_{h}^{k} \frac{| G_{2} (v) |}{| v |_{c}} \leq C h^{k + 1 / 2} (∥ u ∥_{W^{k + 1, \infty} (Γ)} + ∥ u ∥_{H^{k + 1} (^Ω)}) .

Combining the above results we get

sup v \in {˚ W}_{h}^{k} \frac{| G (v_{h}) |}{∥ v_{h} ∥_{a}} \leq C (h^{k} ∥ u ∥_{H^{k + 1} (^Ω)} + h^{4} ∥ u ∥_{W^{4, \infty} (^Ω)}) .

	$\sqrt{h} sup v \in B_{h}^{k} \frac{\| G (v) \|}{\| v \|_{c}} \leq$	$C (h^{7 / 2} ∥ u ∥_{W^{2, \infty} (^Ω)} + h^{5} ∥ u ∥_{W^{4, \infty} (^Ω)})$
		$+ C h^{k + 1 / 2} (∥ u ∥_{W^{k + 1, \infty} (Γ)} + ∥ u ∥_{H^{k + 1} (^Ω)}) .$

The result now follows from Theorem 1.

5. Implementation

One feature of Nitsche’s method, that is preserved with BDT, is that one uses the full space $W_{h}^{k}$ of piecewise polynomials without restriction at the boundary. The modification of

	$\| s_{h} \|_{c}^{2} \leq$	$⎛ ⎝ sup v_{h} \in B_{h}^{k} \frac{\| G (v_{h}) \|}{\| v_{h} \|_{c}} ⎞ ⎠ \| ϕ_{h} \|_{c} + ∥ u_{h} ∥_{a} ∥ ϕ_{h} ∥_{a}$
	$\leq$	$⎛ ⎝ sup v_{h} \in B_{h}^{k} \frac{\| G (v_{h}) \|}{\| v_{h} \|_{c}} ⎞ ⎠ \| s_{h} \|_{c} + c_{1} \sqrt{h} (∥ w_{h} ∥_{a} + c_{1} \sqrt{h} \| s_{h} \|_{c}) \| s_{h} \|_{c} .$

	$∥ w_{h} ∥_{a} \leq$	$⎛ ⎜ ⎝ sup v_{h} \in {˚ W}_{h}^{k} \frac{\| G (v_{h}) \|}{∥ v_{h} ∥_{a}} ⎞ ⎟ ⎠ + ∥ s_{h} ∥_{a}$
		$\leq ⎛ ⎜ ⎝ sup v_{h} \in {˚ W}_{h}^{k} \frac{\| G (v_{h}) \|}{∥ v_{h} ∥_{a}} ⎞ ⎟ ⎠ + c_{1} \sqrt{h} ∥ s_{h} ∥_{c}$
		$\leq ⎛ ⎜ ⎝ sup v_{h} \in {˚ W}_{h}^{k} \frac{\| G (v_{h}) \|}{∥ v_{h} ∥_{a}} ⎞ ⎟ ⎠ + \frac{4}{3} c_{1} \sqrt{h} ⎛ ⎝ sup v_{h} \in B_{h}^{k} \frac{\| G (v_{h}) \|}{\| v_{h} \|_{c}} ⎞ ⎠ + \frac{1}{3} ∥ w_{h} ∥_{a}$

Obtaining higher-order Galerkin accuracy when the boundary is polygonally approximated

Authors

Equal higher order analysis of an unfitted discontinuous Galerkin method for Stokes flow systems

Isogemetric Analysis and Symmetric Galerkin BEM: a 2D numerical study

Variational Time Discretizations of Higher Order and Higher Regularity

Higher order Galerkin-collocation time discretization with Nitsche's method for the Navier-Stokes equations

A new Besse-type relaxation scheme for the numerical approximation of the Schrödinger-Poisson system

Corrected trapezoidal rules for singular implicit boundary integrals

Constructing Nitsche's method for variational problems

1. Introduction

2. The Bramble-Dupont-Thomée approach

2.1. An example of a circle

3. A new method based on a Robin-type approach

Assumption 1.

4. Error Analysis

4.1. Stability Analysis

Lemma 1.

Proof.

Assumption 2.

Theorem 1.

Proof.

Assumption 3.

Theorem 2.

Proof.

5. Implementation

Obtaining higher-order Galerkin accuracy when the boundary is polygonally approximated

Authors

Related Research

Equal higher order analysis of an unfitted discontinuous Galerkin method for Stokes flow systems

Isogemetric Analysis and Symmetric Galerkin BEM: a 2D numerical study

Variational Time Discretizations of Higher Order and Higher Regularity

Higher order Galerkin-collocation time discretization with Nitsche's method for the Navier-Stokes equations

A new Besse-type relaxation scheme for the numerical approximation of the Schrödinger-Poisson system

Corrected trapezoidal rules for singular implicit boundary integrals

Constructing Nitsche's method for variational problems

This week in AI

1. Introduction

2. The Bramble-Dupont-Thomée approach

2.1. An example of a circle

3. A new method based on a Robin-type approach

Assumption 1.

4. Error Analysis

4.1. Stability Analysis

Lemma 1.

Proof.

Assumption 2.

Theorem 1.

Proof.

Assumption 3.

Theorem 2.

Proof.

5. Implementation