Advanced
DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR PARABOLIC PROBLEMS WITH MIXED BOUNDARY CONDITION†
DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR PARABOLIC PROBLEMS WITH MIXED BOUNDARY CONDITION†
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(5_6): 585-598
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : February 25, 2014
  • Accepted : May 24, 2014
  • Published : September 30, 2014
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
MI RAY OHM
HYUN YOUNG LEE
JUN YONG SHIN

Abstract
In this paper we consider the nonlinear parabolic problems with mixed boundary condition. Under comparatively mild conditions of the coefficients related to the problem, we construct the discontinuous Galerkin approximation of the solution to the nonlinear parabolic problem. We discretize spatial variables and construct the finite element spaces consisting of discontinuous piecewise polynomials of which the semidiscrete approximations are composed. We present the proof of the convergence of the semidiscrete approximations in L ( H 1 ) and L ( H 2 ) normed spaces. AMS Mathematics Subject Classification : 65M15, 65N30.
Keywords
1. Introduction
Discontinuous Galerkin (DG) finite element methods employ discontinuous piecewise polynomials to approximate the solutions of differential equations and impose interelement continuity weakly. Even though DG methods often have been involved with large number of degrees of freedom than the classical Galerkin method, DG methods are adopted widely in a variety of differential equations. DG methods were introduced for the numerical solutions of first-order hyperbolic system, but independently they are proposed as nonconforming schemes for the numerical solutions of 2nd order elliptic problems by Nitsche [10] . Recently there has been renewed interest in DG methods due to their efficient properties which include a high degree of locality, the flexibility of locally varying the degree of polynomial in adaptive hp version approximations since no continuity requirement is imposed.
Much attention have been devoted to the analysis of DG methods applied to elliptic problem [6 , 7 , 8] as well as to several other types of nonlinear equations including time-dependent convection-diffusion equations [3] , non-Fickian diffusion equation [14] , Camassa-Holm equation [18] , solid viscoelasticity problems [15] , Maxwell equations [4] , Navier Stokes equations [16] , Keller-Segel chemotaxis model [5] and reactive transport problem [17] .
In this paper we consider the DG methods applied to parabolic problems. In [13] Rieviere and Wheeler initiated to adopt DG method and develop DG approximations to parabolic problems. They constructed discontinuous and time discretized approximations and obtained the optimal convergence order of spatial error in H 1 and time truncation error in L 2 normed space. In [11] the authors applied DG method to parabolic problem with homogeneous Neumann boundary condition and constructed DG spatial discretized approximations using a penalty term and obtained an optimal L ( L 2 ) error estimate. In addition the authors [12] applied DG method to construct the fully discrete approximations for the parabolic problems with homogeneous Neumann boundary condition and obtained the optimal order of convergence in ( L 2 ) normed space.
And also Lasis and Süli [9] considered the hp-version DG method with interior penalty for semilinear parabolic equations to construct spatial discretized approximations and obtained an optimal L ( H 1 ) and L ( L 2 ) error estimates.
In this paper we consider the semidiscrete DG approximations of the nonlinear parabolic equations. Compared to the previous works in this paper we require very weak conditions on the terms characterizing the nonlinearity of the parabolic problem. In this paper we weaken the conditions of the tensor coefficient and the forcing term so that they are assumed to be locally Lipschitz continuous only. In addition, the parabolic problem considered in this paper is related with mixed nonhomogeneous Dirichlet-nonhomogeneous Neumann boundary conditon so that we manage the most generalized boundary condition. The rest of this paper is as follows. In Section 2 we introduce our parabolic problem to be considered and some notations and we construct finite element space. In Section 3, we develop some auxiliary projection onto finite element space and we prove its convergence of optimal order. In Section 4 we construct the semidiscrete approximation and prove its existence and finally we provide the error analysis of the semidiscrete approximations.
2. The problem and notations
Consider the following nonlinear parabolic differential equation:
PPT Slide
Lager Image
where Ω is a bounded open convex domain in ℝ d , 1 ≤ d ≤ 3, Ω is the boundary of Ω, Ω N Ω D = Ω, Ω N Ω D = ϕ and n is a unit outward normal vector to Ω.
Assume that
(A1). a ( x, u ( x, t )) is continuous at
PPT Slide
Lager Image
(A2). There exists a positive constant a * such that a ( x, u ( x, t )) ≥ a * ,
PPT Slide
Lager Image
Let
PPT Slide
Lager Image
be a subdivision of Ω, where Ei is an interval if d = 1, and in case of d = 2( d = 3) Ei is a triangle or a quadrilateral (a symplex or parallelogram) which may have one curved edge (face). Let hi be the diameter of Ei and h = max{ hi : 1 ≤ i Nh }. We assume that there exists a constants δ such that δ −1 h hi δh , 1 ≤ i Nh .
Let εh be the set of the edge of Ei , 1 ≤ i Nh and we let
PPT Slide
Lager Image
where
PPT Slide
Lager Image
is ( d −1) dimensional measure defined in ℝ d−1 . If e = ∂Ei ∂Ej with i < j , the unit outward normal vector ni to Ei is taken as the unit vector n associated with e
The L 2 inner product is denoted by (·, ·) and we denote usual L 2 norm defined on E by ∥ · ∥ E , and usual L norm by ∥ · ∥ ∞,E . In both cases we may skip E if E = Ω. Let Hs ( E ) be the Sobolev space equipped with the usual Sobolev norm
PPT Slide
Lager Image
where
PPT Slide
Lager Image
If E = Ω, we simply denote it by ∥·∥ s and if s = 0 denote it by ∥·∥ E . We denote the usual seminorm defined on E by | · | s,E . And also we denote W s,∞ ( E ) = { v | Dlv L , ∀ | l | ≤ s } equipped with the norm
PPT Slide
Lager Image
If E = Ω then for our convevience we skip E in the notation of W s,∞ ( E ). Now we let Hs h ) = { v | v | Ei Hs ( Ei ) 1 ≤ i Nh }. If v Hs h ) with
PPT Slide
Lager Image
we define the average { v } and the jump ⟨ v ⟩ functions as follows: For e ∂Ei ∂Ej with i < j then
PPT Slide
Lager Image
For e Ω D ,
PPT Slide
Lager Image
Now we define the following broken norm on H 2 h ):
PPT Slide
Lager Image
To continue our analysis we may assume that Ei is a triangle. For the case that Ei is a rectangle we may develop the analogous theories. We let
PPT Slide
Lager Image
be the space of piecewise polynomials defined as
PPT Slide
Lager Image
where Pr ( Ej ) is the set of polynomials of total degree ≤ r .
3. Approximation properties and elliptic projection
Hereafter C denotes a generic positive constant independent of h and any two C s in different positions don’t need to be the same. The following approximation properties are proved in [1 , 2] .
Lemma 3.1. Let E ∈ Ω h , e be an edge of E and v Hs ( E ). Then there exist a positive constant independent of v, r and h and a sequence
PPT Slide
Lager Image
r = 1, 2, · · · , such that for any 0 ≤ q s ,
PPT Slide
Lager Image
hold where μ = min( r + 1, s ). Moreover if e = ∂Ei ∂Ej then
PPT Slide
Lager Image
holds .
Lemma 3.2. Let E ∈ Ω h , e be an edge of E and n be a normal vector associated with e. Then there exists a positive constant C such that v H 1 ( E )
PPT Slide
Lager Image
Now we let
PPT Slide
Lager Image
be the interpolation of u satisfying the approximation properties of Lemma 3.1. By applying Lemma 3.1 we obviously have the following Lemma.
Lemma 3.3. If u Hs (Ω), then
PPT Slide
Lager Image
satisfies the following approximation property
PPT Slide
Lager Image
where μ = min( r + 1, s ).
Proof . By applying Lemma 3.1 and 3.2, we get
PPT Slide
Lager Image
We define the following bilinear form Aβ ( a, u : v,w ) on
PPT Slide
Lager Image
PPT Slide
Lager Image
For a α > 0 we let
PPT Slide
Lager Image
Lemma 3.4. For any v,w Hs h ) with s ≥ 2
PPT Slide
Lager Image
holds.
Proof . For any v,w Hs h ) with s ≥ 2,
PPT Slide
Lager Image
Lemma 3.5. If β is sufficiently large, then there is a constant
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
Proof . For
PPT Slide
Lager Image
and δ > 0
PPT Slide
Lager Image
holds. By applying Lemma 3.2 we get for sufficiently large β and δ ,
PPT Slide
Lager Image
since
PPT Slide
Lager Image
By applying Lemmas 3.4 and 3.5 there exists
PPT Slide
Lager Image
satisfying
PPT Slide
Lager Image
By Lax-Milgram Lemma, ũ satisfies
PPT Slide
Lager Image
Now we let
PPT Slide
Lager Image
Lemma 3.6. Let G be a linear mappping defined on H 2 h ) and suppose that there exists w H 2 h ) satisfying
PPT Slide
Lager Image
Suppose that there exist positive constants K 1 and K 2 such that
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Then we have the following estimation
PPT Slide
Lager Image
Proof . Let ϕ be the solution of
PPT Slide
Lager Image
Then by the regularity property of elliptic problem we get ∥ ϕ 2 C w ∥. By Lemma 3.3 there exists
PPT Slide
Lager Image
an interpolation ϕ satisfying
PPT Slide
Lager Image
By (3.3) we have
PPT Slide
Lager Image
which implies that
PPT Slide
Lager Image
Theorem 3.1. Suppose that u (·, t ) ∈ Hs , ut (·, t ) ∈ Hs then the following error estimations hold:
PPT Slide
Lager Image
where μ = min( r +1, s ). And also if
PPT Slide
Lager Image
are bounded for e εh .
Proof . By Lemma 3.4 and 3.5, we have
PPT Slide
Lager Image
so that
PPT Slide
Lager Image
By (3.1), (3.2) and Lemma 3.6 with G ( v ) = 0, we get
PPT Slide
Lager Image
Now we differentiate
PPT Slide
Lager Image
with respect to t to obtain
PPT Slide
Lager Image
where
PPT Slide
Lager Image
By the similar process as the proof of Lemma 3.4 we have with v H 2 h ),
PPT Slide
Lager Image
and with
PPT Slide
Lager Image
PPT Slide
Lager Image
Therefore, by Lemma 3.6
PPT Slide
Lager Image
By Lemma 3.5, Lemma 3.4 and (3.6) we have
PPT Slide
Lager Image
which implies that by Lemma 3.3,
PPT Slide
Lager Image
Hence we get
PPT Slide
Lager Image
Now we substitute (3.8), (3.2) and (3.5) into (3.7) to get
PPT Slide
Lager Image
If
PPT Slide
Lager Image
then
PPT Slide
Lager Image
holds. Now we let e = Ei Ej . By Lemma 3.1
PPT Slide
Lager Image
holds.
4. Spatial discretized approximation and error analysis.
The discontinuous Galerkin method of the problem (1.1) reads as follows: find
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
where
PPT Slide
Lager Image
and Ph ( u 0 ( x )) denotes the approximation of u 0 ( x ) generated by Lemma 3.1. From (2.1) u ( x, t ) satisfies
PPT Slide
Lager Image
Theorem 4.1. There exists uh ( x, t ) satisfying (4.1). If f ( x, t, u ) and a ( x, u ) are locally Lipschitz continuous in t and u then there exists a unique uh ( x, t ) locally. And also if f ( x, t, u ) and a ( x, u ) are globally Lipschitz continuous in t and u then the unique existence holds globally .
Proof . Let
PPT Slide
Lager Image
be a basis of
PPT Slide
Lager Image
and suppose that
PPT Slide
Lager Image
From (4.1) we have
PPT Slide
Lager Image
Let α ( t ) = ( α 1 ( t ), α 2 ( t ), · · · , αm ( t )) T . (4.3) can be represented as the following system
PPT Slide
Lager Image
where M = ( Mij ) 1≤i,jm , N ( α ( t )) = ( Nij ( α )) 1≤i,jm are symmetric matrices and F ( α ( t )) = Fj ( α ( t )) 1≤jm and L ( t ) = ( Lj ( t )) 1≤jm are vectors. M,N ( α ( t )) and L ( t ) are defined by Mij = ( vi ( x ), vj (x)), Nij ( α ) = Aβ ( a , uh : vi ( x ), vj ( x )) and Fj ( α ( t )) = ( f ( x, t, uh ), vj ( x )), Lj ( t ) = lβ vj ( x )).
For y = ( y 1 , y 2 , · · · , ym ) T ∈ ℝ m we let
PPT Slide
Lager Image
then
PPT Slide
Lager Image
Therefore M is a positive definite matrix. By applying the theory on the existence of the solution of the system of the ordinary differential equations, we acquire the existence of the solution of the system (4.3).
Since f ( x, t, u ) and a ( x, u ) are locally Lipschitz continuous in u , − N ( α ( t )) α ( t )+ F ( α ( t )) + L ( α ( t )) is also locally Lipschitz in α ( t ). Thus from the theory on the uniqueness property of the system of the ordinary differential equations the unique existence can be quaranteed locally at (0, α (0)) T .
By the similar analysis we may prove that the uniqueness property of uh ( x, t ) holds globally if a ( x, t ) and f ( x, t, u ) are globally Lipschitz in t and u .
Remark 1. From theorem 4.1 we obviously deduce that ∥ uh ( t )∥ L∞ is continuous with respect to t and
PPT Slide
Lager Image
holds for some positive constant K * and sufficiently small h provided that μ = min( r + 1, s ) ≥ 1. We define K * satisfying (4.11) which appears in the end this paper as well as (4.4).
Now we let
PPT Slide
Lager Image
then u uh = η + χ .
Theorem 4.2. We assume that the hypotheses of Theorem 3.1 hold. Suppose that f ( x, t, u ) and a ( x, u ) satisfy that
PPT Slide
Lager Image
hold. If
PPT Slide
Lager Image
then there is a generic positive constant C such that
PPT Slide
Lager Image
where μ = min( r + 1, s ).
Proof . To get the error bound of u uh we temporarily assume that
PPT Slide
Lager Image
holds for sufficiently small h * . We subtract (4.1) from (4.2) and obtain the following error equation:
PPT Slide
Lager Image
from which we have
PPT Slide
Lager Image
By (3.1) we have
PPT Slide
Lager Image
By (4.5), (4.6), Lemma 3.1 and Theorem 3.1 we have for ϵ > 0,
PPT Slide
Lager Image
By Theorem 3.1 we get the following estimations:
PPT Slide
Lager Image
PPT Slide
Lager Image
Therefore
PPT Slide
Lager Image
where C depends on
PPT Slide
Lager Image
and ∥ u s . By the assumption (4.5) and (4.6) we get
PPT Slide
Lager Image
Now we substitute the estimations of L 1 L 3 and (4.8) into (4.7) with v = χ we get for some
PPT Slide
Lager Image
PPT Slide
Lager Image
Now we choose sufficiently small ϵ > 0 and apply the Gronwall inequality to get
PPT Slide
Lager Image
Therefore
PPT Slide
Lager Image
Now we will verify that we may without loss of generality assume that (4.6) holds.
By (4.4), obviously (4.6) holds for t = 0. Suppose that there exist t * such that ∥ u ( t ) − uh ( t )∥ L∞ < 2 K * , ∀ t < t * but
PPT Slide
Lager Image
Now we choose a sequence of
PPT Slide
Lager Image
converging to t * . By following the preceding process below (4.6) we obtain the result ∥ χ ( tn )∥ 2 Ch . By applying Lemma 3.1, (3.4) and (4.9) we get
PPT Slide
Lager Image
provided that
PPT Slide
Lager Image
By Theorem 4.1 we notice that uh ( t ) is continuous with respect to t , therefore this implies that ∥( u uh )( t )∥ is continuous with respect to t . Hence we get
PPT Slide
Lager Image
which contradicts to (4.10). Therefore we may assume that (4.6) holds for any h < h * with sufficiently small h * .
BIO
Mi Ray Ohm received her BS degree from Busan National University and Ph.D degree from Busan National University under the direction of Professor Ki Sik Ha. She is a professor at Dongseo University. Her research is centered numerical analysis methods on partial differential equations.
Division of Information Systems Engineering, Dongseo University, 617-716, Korea
e-mail : mrohm@dongseo.ac.kr
Hyun Yong Lee received her BS degree from Busan National University and Ph.D degree from University of Tennessee under the direction of Professor Ohannes Karakashian. She is a professor at Kyungsung University. Her research is centered numerical analysis methods on partial differential equations.
Department of Mathematics, Kyungsung University, 608-736, Korea
e-mail : hylee@ks.ac.kr
Jun Yong Shin received his BS degree from Busan National University and Ph.D degree from University of Texas under the direction of Professor R. Kannau. He is a professor at Pukyong National University. His research is centered numerical analysis methods on partial differential equations.
Department of Applied Mathematics, Pukyong National University, 608-737, Korea
e-mail : jyshin@pknu.ac.kr
References
Babuška I. , Suri M. (1987) The h-p version of the finite element method with quasi-uniform meshes RAIRO Model. Math. Anal. Numer. 21 199 - 238
Babuška I. , Suri M. (1987) The optimal convergence rates of the p-version of the finite element method SIAM J. Numer. Anal. 24 750 - 776    DOI : 10.1137/0724049
Cockburn B. , Shu C-W. (1998) The local discontinuous Galerkin method for time-dependent convection-diffusion systems SIAM J. Numer. Anal. 35 (6) 2440 - 2463    DOI : 10.1137/S0036142997316712
Descombes S. , Lanteri S. (2013) Locally implicit time integration strategies in a discontinuous Galerkin method for Maxwell’s equations J. Sci. Comput. 56 190 - 218    DOI : 10.1007/s10915-012-9669-5
Epshteyn Y. , Kurganov A. (2008) New interior penalty discontinuous Galerkin methods for the Keller-Segel Chemotaxis model SIAM J. Numer. Anal. 47 (1) 386 - 408    DOI : 10.1137/07070423X
Hufford C. , Xing Y. (2014) Superconvergence of the local discontinuous Galerkin method for the linearized Korteweg-de Vries equation J. Compu. Appl. Math. 255 441 - 455    DOI : 10.1016/j.cam.2013.06.004
Johanson A. , Larson M. (2013) A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary Numer. math. 123 607 - 628    DOI : 10.1007/s00211-012-0497-1
Karakashian O. , Pascal F. (2003) A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems SIAM. J. Numer. Anal. 41 2374 - 2399    DOI : 10.1137/S0036142902405217
Lasis A. , Süli E. (2007) hp-version discontinuous Galerkin finite element method for semilinear parabolic problems SIAM J. Numer. Anal. 45 (4) 1544 - 1569    DOI : 10.1137/050642125
Nitsche J. (1971) Über ein Variationspringzip zvr Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36 9 - 15    DOI : 10.1007/BF02995904
Ohm M.R. , Lee H.Y. , Shin J.Y. (2006) Error estimates for a discontinuous Galerkin method for nonlinear parabolic equations Journal of Math. Anal. and Appli. 315 132 - 143    DOI : 10.1016/j.jmaa.2005.07.027
Ohm M.R. , Lee H.Y. , Shin J.Y. (2010) Error estimates for fully discrete discontinuous Galerkin method for nonlinear parabolic equations Journal of Applied Mathematics and Informatics 28 (3-4) 953 - 966
Rivière B. , Wheeler M.F. 2000 A discontinuous Galerkin method applied to nonlinear parabolic equations, In: B. Cockburn, G. E. Karaniadakis, C.-W. Shu (Eds.), Discontinuous Galerkin Method: Theory, Computation and Applications, in: Lecture notes in comput. sci. engrg. vol. 11 Springer Berlin 231 - 244
Rivière B. , Shaw S. (2006) Discontinuous Galerkin finite element approximation of nonlinear non-fickian diffusion in viscoelastic polymers SIAM J. Numer. Anal. 44 (6) 2650 - 2670    DOI : 10.1137/05064480X
Rivière B. , Shaw S. , Whiteman J.R. (2007) Whiteman Discontinuous Galerkin finite element methods for dynamic linear solid viscoelasticity problems Numer. Meth. for Partial Diff. Equa. 23 1149 - 1166    DOI : 10.1002/num.20215
Rhebergen S. , Cockburn B. , Vegt J. (2013) A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations J. Comp. Physics 233 339 - 358    DOI : 10.1016/j.jcp.2012.08.052
Sun S. , Wheeler M. (2005) Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media SIAM J. Numer. Anal. 43 (1) 195 - 219    DOI : 10.1137/S003614290241708X
Xu Y. , Shu C.-W. (2008) A local discontinuous Galerkin method for the Camassa-Holm equation SIAM J. Numer. Anal. 46 (4) 1998 - 2021    DOI : 10.1137/070679764