Galerkin alternating-direction methods for a kind of nonlinear hyperbolic equations on nonrectangular regions

doi:10.1016/j.amc.2006.09.018

Applied Mathematics and Computation

Volume 187, Issue 2, 15 April 2007, Pages 1063-1075

https://doi.org/10.1016/j.amc.2006.09.018 Get rights and content

Abstract

Patch approximation and an approximation to the Jacobian of the isoparametric map are applied to discuss a three-level and a four-level Galerkin alternating-direction procedures for a kind of nonlinear hyperbolic equation which is given on a nonrectangular, curved region. With these procedures, multi-dimensional problems can be solved as a series of one-dimensional problems. And the L² norm error estimates are derived.

Introduction

Galerkin alternating-direction procedure, which was first proposed by Douglas and Dupont [1], is designed to offer a significant reduction in the computing time, storage requirements and to have high accuracy. Since then, a considerable amount of research has been focused on this method, see [2], [3], [4], [5], [6], [7], [8] for instance. Dendy, Fairweather, Hayes, Bramble, Ewing, Cui, Liu and Chen have given further investigation. However their research was restricted to the parabolic or hyperbolic problems on regions of rectangle or unions of rectangles. This is not applicable to many practical problems, see [9], [10] for example. So Hayes developed a new method to solve the parabolic problems on curved regions by using patch approximation and an approximation to the Jacobian of the isoparametric map, see [11], [12], [13]. The work about Galerkin alternating-direction method of nonlinear hyperbolic problems on curved regions has not been found in relative literatures.

In this paper, based on the Galerkin alternating-direction method and patch approximation, a nonlinear hyperbolic problem on curved regions $\tilde{q} (ξ, u) u_{tt} - \sum_{i, j = 1}^{N} \frac{\partial}{\partial ξ_{i}} ({\tilde{a}}_{ij} (ξ, u) \frac{\partial u}{\partial ξ_{j}}) + \sum_{i = 1}^{N} {\tilde{b}}_{i} (ξ, u) \frac{\partial u}{\partial ξ_{i}} = f (ξ, t, u), (ξ, t) \in Ω_{g} \times J,$ $u (ξ, t) = 0, (ξ, t) \in \partial Ω_{g} \times J,$ $u (ξ, 0) = u_{0} (ξ), ξ \in Ω_{g},$ $u_{t} (ξ, 0) = u_{1} (ξ), ξ \in Ω_{g}$ is considered, where Ω_g is a bounded domain in R^N(N ⩾ 2), ξ = (ξ₁, ξ₂, … , ξ_N) ∈ R^N, J = (0, T]. ∂Ω_g, the boundary of Ω_g, is piecewise polynomial.

An outline of the paper is as follows. In Section 2, some notations and assumptions are introduced. In Section 3, a three-level and a four-level Galerkin alternating-direction methods are proposed. In Section 4, the matrix problems and the start-up procedures of the above two methods are given. Finally, in Section 5 and in Section 6, the L² norm error estimates of the two procedures are derived respectively by using the theory and techniques of priori estimate of differential equations. The result of this paper is important for the theoretical analysis and practical computation of a kind of nonlinear vibration problem.

Section snippets

Preliminary and notations

Let Ω_g be the given curved region with coordinates ξ = (ξ₁, ξ₂, … , ξ_N) ∈ R^N, and let Ω be a master rectangular regions with coordinates x = (x₁, x₂, … , x_N) ∈ R^N. Define $(f, g)_{Ω_{g}} = \int_{Ω_{g}} fg d ξ, (f, g) = \int_{Ω} fg d x$ and let $ρ (x) = ∣ J (F^{- 1}) ∣ = \frac{\partial (ξ)}{\partial (x)}$ , where the map F is an invertible map from Ω_g to Ω, ∣J(F⁻¹)∣ is the determinant of the Jacobian of F⁻¹. If the regular families of quadrilateral isoparametric elements (see [14], [15] for instance) is used to define F, there exist positive constants ρ_∗ and ρ^∗ such that ρ_∗ ⩽ ρ(x) ⩽ ρ^∗ for

Formulation of Galerkin alternating-direction method

The weak form of problem (1.1), (1.2), (1.3), (1.4) is $\{\begin{matrix} (\tilde{q} (ξ, u) u_{tt}, v)_{Ω_{g}} + \sum_{i, j = 1}^{N} {({\tilde{a}}_{ij} (ξ, u) \frac{\partial u}{\partial ξ_{j}}, \frac{\partial v}{\partial ξ_{i}})}_{Ω_{g}} + \sum_{i = 1}^{N} {({\tilde{b}}_{i} (ξ, u) \frac{\partial u}{\partial ξ_{i}}, v)}_{Ω_{g}} = (f (ξ, t, u), v)_{Ω_{g}}, \\ v \in H_{0}^{1} (Ω_{g}), t \in (0, T], \\ (u (ξ, 0), v)_{Ω_{g}} = (u_{0}, v)_{Ω_{g}}, \\ (u_{t} (ξ, 0), v)_{Ω_{g}} = (u_{1}, v)_{Ω_{g}} . \end{matrix}$ On the master region Ω, (3.1) becomes $\{\begin{matrix} (ρ q (u) u_{tt}, v) + \sum_{i, j = 1}^{N} (ρ a_{ij} (u) \frac{\partial u}{\partial x_{j}}, \frac{\partial v}{\partial x_{i}}) + \sum_{i = 1}^{N} (ρ b_{i} (u) \frac{\partial u}{\partial x_{i}}, v) = (ρ f, v), v \in H_{0}^{1} (Ω_{g}), t \in (0, T], \\ (ρ u (x, 0), v) = (ρ u_{0}, v) = (u_{0}, v)_{Ω_{g}}, (ρ u_{t} (x, 0), v) = (ρ u_{1}, v) = (u_{1}, v)_{Ω_{g}} . \end{matrix}$ The modified Laplace Galerkin alternating-direction method of (3.2) can be defined as $(ρ q (U^{n}) \partial^{2} U^{n}, V) + λ (ρ q (U^{n}) \nabla (U^{n + 1} - 2 U^{n} +$

Matrix problems

There are two orderings of the nodes in the master region Ω, see [11]. Denote S_h(Ω) = span{N_i}, the tensor product basis is $N_{i} (x) = γ_{p (i)}^{1} (x) γ_{h (i)}^{2} (y) = γ_{p}^{1} (x) γ_{h}^{2} (y)$ , x = (x, y), where p(i) is the grid line number in x-direction of global node i, h(i) is the grid line number in y-direction of global node i, p = 1, 2, … , N_x, h = 1, 2, … , N_y, m = N_xN_y, i = 1, 2, … , m.

Denote Ω_i = supp(N_i), Ω_i,j = supp(N_i) ∩ supp(N_j). Let $\tilde{ρ} = {{\tilde{ρ}}_{ij}}_{i, j = 1}^{m}$ be the approximation of ρ(x) and $q^{n} = {q_{ij}^{n}}$ be the approximation of q(Uⁿ). Because ρ(x) is

Priori error estimates of three-level scheme

Let ξⁿ = Uⁿ − Wⁿ, ηⁿ = Wⁿ − uⁿ for V ∈ S_h(Ω), n ⩾ 1, (2.5), (3.2), (3.4) yields $\sum_{i = 1}^{3} L_{i}^{n} = (ρ q (U^{n}) \partial^{2} ξ^{n}, V) + λ (ρ q (U^{n}) \nabla (ξ^{n + 1} - 2 ξ^{n} + ξ^{n - 1}), \nabla V) + λ^{2} (Δ t)^{4} (\tilde{ρ} q^{n} \frac{\partial^{2}}{\partial x \partial y} \partial^{2} ξ^{n}, \frac{\partial^{2} V}{\partial x \partial y}) = ((ρ q (U^{n}) - \tilde{ρ} q^{n}) \partial^{2} ξ^{n}, V) + λ ((ρ q (U^{n}) - \tilde{ρ} q^{n}) \nabla (ξ^{n + 1} - 2 ξ^{n} + ξ^{n - 1}), \nabla V) + (I_{1}^{n}, V) + (I_{2}^{n}, V) + (I_{3}^{n}, \nabla V) + (I_{4}^{n}, \frac{\partial^{2} V}{\partial x \partial y}),$ where $\begin{matrix} I_{1}^{n} = ρ μ η^{n + 1} - ρ q (U^{n}) \partial^{2} W^{n} + ρ q (u^{n + 1}) u_{tt}^{n + 1} + ρ (f (U^{n}) - f (u^{n + 1})) + ρ b (u^{n + 1}) \nabla W^{n + 1} - ρ b (U^{n}) \nabla U^{n}, \\ I_{2}^{n} = (ρ q (U^{n}) - \tilde{ρ} q^{n}) \partial^{2} W^{n}, \\ I_{3}^{n} = ρ a (u^{n + 1}) \nabla W^{n + 1} - ρ a (U^{n}) \nabla U^{n} - λ \tilde{ρ} q^{n} \nabla (W^{n + 1} - 2 W^{n} + W^{n - 1}), \\ I_{4}^{n} = - λ^{2} (Δ t)^{4} \tilde{ρ} q^{n} \frac{\partial^{2}}{\partial x \partial y} \partial^{2} W^{n} . \end{matrix}$ With V = ξⁿ⁺¹ − ξⁿ⁻¹ = Δt(d_tξⁿ + d_tξⁿ⁻¹) as the test function in (5.1), and note

Priori error estimates of four-level scheme

Let ξⁿ = Uⁿ − Wⁿ, ηⁿ = Wⁿ − uⁿ, forV ∈ S_h(Ω), n ⩾ 1, (2.5), (3.2), (3.5) yields $(ρ q (U^{n}) \partial^{2} ξ^{n}, V) + λ (ρ q (U^{n}) \nabla (ξ^{n + 1} - 2 ξ^{n} + ξ^{n - 1}), \nabla V) + λ^{2} (Δ t)^{4} (\tilde{ρ} q^{n} \frac{\partial^{2}}{\partial x \partial y} \partial^{2} ξ^{n}, \frac{\partial^{2} V}{\partial x \partial y}) = ((ρ q (U^{n}) - \tilde{ρ} q^{n}) (\partial^{2} ξ^{n} - \partial^{2} ξ^{n - 1}), V) + λ ((ρ q (U^{n}) - \tilde{ρ} q^{n}) \nabla (ξ^{n + 1} - 2 ξ^{n} + ξ^{n - 1}), \nabla V) + (I_{1}^{n}, V) + ({\tilde{I}}_{2}^{n}, V) + (I_{3}^{n}, \nabla V) + (I_{4}^{n}, \frac{\partial^{2} V}{\partial x \partial y}),$ where $I_{1}^{n}$ , $I_{3}^{n}$ and $I_{4}^{n}$ are defined as above in (5.1), and ${\tilde{I}}_{2}^{n} = (ρ q (U^{n}) - \tilde{ρ} q^{n}) (\partial^{2} W^{n} - \partial^{2} W^{n - 1}) .$ All of the terms in (6.1) occurred in (5.1) and are handled exactly as in Theorem 1 except

(a)
$((ρ q (U^{n}) - \tilde{ρ} q^{n}) (\partial^{2} ξ^{n} - \partial^{2} ξ^{n - 1}), V)$ ,
(b)
$({\tilde{I}}_{2}^{n}, V) = ((ρ q (U^{n}) - \tilde{ρ} q^{n}) (\partial^{2} W^{n} - \partial^{2} W^{n - 1}), V)$ ,

Acknowledgements

This work was supported by the Major State Basic Research Program of China (Grant No. 1990328), the National Natural Science Foundation of China (Grant Nos. 10372052 and 10271066) and the Directorate Foundation of the State Education Commission (Grant No. 20030422047).

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Applied Mathematics and Computation

Galerkin alternating-direction methods for a kind of nonlinear hyperbolic equations on nonrectangular regions

Abstract

Introduction

Section snippets

Preliminary and notations

Formulation of Galerkin alternating-direction method

Matrix problems

Priori error estimates of three-level scheme

Priori error estimates of four-level scheme

Acknowledgements

Alternating direction Galerkin method on rectangles

Alternating-Galerkin methods for parabolic and hyperbolic problems on rectangular polygon

SIAM J. Numer. Anal.

An alternating-direction Galerkin method for a class of second-order hyperbolic equation in two space variables

SIAM J. Numer. Anal.

Alternating-direction along flow lines in fluid flow problem

Comput. Methods Appl. Mech. Eng.

Alternating direction multistep methods for parabolic problem-iterative stabilization

SIAM J. Numer. Anal.