An exponentially graded mesh for singularly perturbed problems Christos Xenophontos Department of Mathematics and Statistics University of Cyprus joint work with P. Constantinou (UCY), S. Franz (TU Dresden) and L. Ludwig (TU Dresden)
Introduction Let u ( x) exp( x / ), x (0,1), R. BL 2
Introduction Let u ( x) exp( x / ), x (0,1), R. BL We want to construct a mesh such that the quantity u BL u is minimized, where u is the finite element approximation and is some norm (e.g. L 2, H 1 ). 2
The mesh points m xii 0 grading function Γ(x), via will be defined in terms of a i ( xi ), i 0,..., m. m 3
The mesh points m xii 0 grading function Γ(x), via will be defined in terms of a i ( xi ), i 0,..., m. m The function Γ(x) should satisfy (0,1) [0,1], ( ) 0 in (0,1), (0) 0, (1) 1. 1 0 C C x 3
The mesh points m xii 0 grading function Γ(x), via will be defined in terms of a i ( xi ), i 0,..., m. m The function Γ(x) should satisfy (0,1) [0,1], ( ) 0 in (0,1), (0) 0, (1) 1. 1 0 C C x (ote that a uniform mesh corresponds to Γ(x) = x.) 3
We next define where Π p denotes the space of polynomials of degree p and V u H ( I) : u ( I ), j 1,..., m 1 0 I p j I,, 1,...,. j x j1 x j j m j 4
We next define where Π p denotes the space of polynomials of degree p and V u H ( I) : u ( I ), j 1,..., m 1 0 I p j I,, 1,...,. j x j1 x j j m There holds [Xenophontos, 1996] BL 2 L p BL p 2 j u w A ( ), ( u w ) A ( ) where w V and p p p1 p,, L 1/2 1 2 p 2 x/ Ap, ( ) ( x) e dx. 0 4
By considering all such grading functions, we find the optimal grading function Γ * that minimizes A p,ε (Γ) as 5 *,, 1 ( ) 1 exp, ( 1) p p x x C p, 1 exp. ( 1) C p p R
By considering all such grading functions, we find the optimal grading function Γ * that minimizes A p,ε (Γ) as * 1 x p, ( x) 1 exp, C p, ( p1) C p, 1 exp. ( p 1) R The optimal mesh points are obtained from 1 j j p,, 0,..., m * * x j m 5
C x j p j j m m p, ( 1)ln 1, 0,.... 6
C x j p j j m m p, ( 1)ln 1, 0,.... 0 1 6
C x j p j j m m p, ( 1)ln 1, 0,.... 0 1 0 x /2 1 1 6
C x j p j j m m p, ( 1)ln 1, 0,.... 0 1 Equidistant mesh with /2 intervals ( > 2 even) 0 x /2 1 1 6
C x j p j j m m p, ( 1)ln 1, 0,.... 0 1 Equidistant mesh with /2 intervals ( > 2 even) 0 x /2 1 1 The mesh is given by a continuous, monotonically increasing, piecewise continuously differentiable generating function φ s.t. φ(0) = 0 [Roos and Linß (1999)]. 6
The mesh points are given by ( p 1) ( i / ), i 0,1,..., / 2 1 xi 1 x /21 x /21 i / 2 1, i / 2,...,1 / 2 1 where ( t) ln 1 2 C p, t, t [0,1/ 2 1/ ] C p, 1 exp. ( p 1) R 7
We note that the meshwidth in the interval [0, x /2 1 ] satisfies 8
We note that the meshwidth in the interval [0, x /2 1 ] satisfies h j x j x j1 ( p 1) ( j / ) (( j 1) / ) ( p 1) max C ( p 1) I j C 1 1 8
We note that the meshwidth in the interval [0, x /2 1 ] satisfies Also h j x j x j1 ( p 1) ( j / ) (( j 1) / ) ( p 1) max C ( p 1) I j C 1 1 e e e x /2 1 / ( p1) (( /21)/ ) p, e ( p1) C ln 1/ ( p1) ( p1)ln 1 2 C ( /21)/ 8
Lemma: Let u I be the degree p piecewise interpolant of u BL defined on the exponential mesh. Then I ubl u C L ( I) I ubl u C 2 L ( I) ( p1) ( p1) I 1/2 p ( ubl u ) C. 2 L ( I),, 9
A model Reaction-Diffusion problem Find ux ( ) such that where ( ) ( ) ( ) ( ) in (0,1) u(0) 0, u(1) 0 2 u x a x u x f x I 2 0 1, a( x) 0, f ( x) L ( I) are given. 10
A model Reaction-Diffusion problem Find ux ( ) such that where ( ) ( ) ( ) ( ) in (0,1) u(0) 0, u(1) 0 2 u x a x u x f x I 2 0 1, a( x) 0, f ( x) L ( I) are given. The solution may be decomposed as u us ubl where u ( x) C, u ( x) C e, k 0,1,2,... ( k ) ( k ) k x/ S BL 10
Variational Formulation Find u H 1 0 (I) such that where B( u, v) F( v) v H ( I) 1 0 2 B( u, v) u ( x) v ( x) a( x) u( x) v( x) dx I F( v) f ( x) v( x) dx I The natural energy norm is defined as u E B( u, u) 2 11
As usual, we seek and we have Discretization u V H I, 1 0 s. t. B u v F v v V u u v u v V E E 12
As usual, we seek and we have Discretization u V H I, 1 0 s. t. B u v F v v V u u v u v V E Theorem: Let u be the finite element solution to u, E based on the exponential mesh. Then 1/2 u u 2 u u L ( I) C 2 L ( I) u u C L ( I) p p 12
Remark: The above result in balanced in the following sense: 13
Remark: The above result in balanced in the following sense: u u u O 1/2 BL ( ) E BL 0, I BL 0, I 13
Remark: The above result in balanced in the following sense: u u u O 1/2 BL ( ) E BL 0, I BL 0, I u u O BL 1/2 (1) 0, I BL 0, I 13
Remark: The above result in balanced in the following sense: u u u O 1/2 BL ( ) E BL 0, I BL 0, I u u O BL 1/2 (1) 0, I BL 0, I As ε 0, the energy norm does not see the layer. In contrast, the balanced norm does. 13
A model Convection-Diffusion problem Find ux ( ) such that u ( x) b( x) u( x) c( x) u( x) f ( x) in I (0,1) u(0) 0, u(1) 0 2 where b x c x L I f x L I 0 1,1< ( ),0 ( ) ( ), ( ) ( ) are given. 14
A model Convection-Diffusion problem Find ux ( ) such that u ( x) b( x) u( x) c( x) u( x) f ( x) in I (0,1) u(0) 0, u(1) 0 2 where b x c x L I f x L I 0 1,1< ( ),0 ( ) ( ), ( ) ( ) are given. The solution may be decomposed as where u u u S BL u ( x) C, u ( x) C e, k 0,1,2,... ( k ) ( k ) k x/ S BL 14
Variational Formulation Find u H 1 0 (I) such that where B( u, v) F( v) v H ( I) 1 0 B( u, v) u( x) v( x) b( x) u( x) c( x) u( x) v( x) dx I F( v) f ( x) v( x) dx I The natural energy norm is defined as (and it is balanced). u E B( u, u) 1/2 15
As usual, we seek and we have Discretization u V H I 1 0 s. t. B u, v F v v V B( u u, v) 0 v V 2 1 0 B( u, u) C u u H I E 16
As usual, we seek and we have Discretization u V H I 1 0 s. t. B u, v F v v V B( u u, v) 0 v V 2 1 0 B( u, u) C u u H I E Theorem: Let u be the finite element solution to u, based on the exponential mesh Then u u C E p 16
The exponential mesh in 2D The boundary layer phenomenon is one-dimensional (namely in the direction normal to the boundary). Usually u g e R (, ) /, BL where (ρ, θ) are boundary fitted coordinates, representing the distance to the boundary and the arc length, respectively, and g(θ) is a smooth function. 17
Mesh Design near Ω (smooth domain) 18
Mesh Design near Ω (smooth domain) Mesh Design (square) 18
The one-dimensional results carry over to the twodimensional case, almost verbatim. 19
The one-dimensional results carry over to the twodimensional case, almost verbatim. Reaction-Diffusion: in smooth domains u cu f in u 0 on R 2 2 19
The one-dimensional results carry over to the twodimensional case, almost verbatim. Reaction-Diffusion: in smooth domains u cu f in u 0 on R 2 2 Convection-Diffusion: T u b u cu f in [0,1] u 0 on T with b b ( x, y), b ( x, y) [, ] 0 1 2 1 1 2 19
In both cases we are able to show u u C FEM L ( ) p where is the number of subdivisions in the exponential mesh (in one-direction). Also, for reaction-diffusion u u 1/2 FEM 2 u u C 1 L ( ) FEM H ( ) p and for convection-diffusion u u C FEM E p 20
umerical Results We consider the problem u u u u f in [0,1] x y u 0 on 2 2 with f chosen so that (1 x)/ 1/ 2(1 y)/ 2/ ( e e ) ( e e ) u sin( x / 2) y 1/ 2/ 1e 1e 21
FEM convergence using the exponential mesh 22
We also compare the exponential mesh with the Bakhvalov- Shishkin mesh and with a mesh by Roos-Teofanov-Uzelac (both layer adapted, optimally convergent) 23
We also compare the exponential mesh with the Bakhvalov- Shishkin mesh and with a mesh by Roos-Teofanov-Uzelac (both layer adapted, optimally convergent) 23
Closing Remarks The proposed exponential mesh is well suited for singularly perturbed problems in one- and twodimensions. The proof of robustness and optimal convergence is achieved due to the results of Roos & Linß (1999). umerical experiments indicate that the proposed method (slightly) outperforms other equally robust and uniformly convergent methods. 24