Robust exponential convergence of hp-fem for singularly perturbed systems of reaction-diffusion equations Christos Xenophontos Department of Mathematics and Statistics University of Cyprus joint work with J. M. Melenk (TU Wien) and L. Oberbroeckling (Loyola Univ. MD)
The Model Problem Find U( x) u( x), v( x) such that T ( ) ( ) ( ) ( ) in (0,1) U(0) U(1) 0, E U x A x U x F x I where E, 2 0 2,0 1, 0 and A a ( x) a ( x) f( x) a21( x) a22( x) gx ( ) 11 12 ( x), F( x) are given. 2
Assumptions: (1) The functions a ij (x), f (x),g(x) are analytic, and C, C,,,, C, f g f g a a ( n) n f C, I f fn n ( n) n g C, I g gn n ( n) n ij, I a a!, 0 0 0!, a C n! n, i, j 1,2 (2) The matrix A is pointwise positive definite, i.e. fixed > 0, such that T T 2 ξ Aξ ξ ξ ξ x I 3
Remark: Both components of the solution have boundary layers at x = 0 and x = 1, of width O(ln), but the second component has an additional sublayer of width O(ln). 4
Remark: Both components of the solution have boundary layers at x = 0 and x = 1, of width O(ln), but the second component has an additional sublayer of width O(ln). This is illustrated in the figure below, which shows the solution corresponding to 2 1 1 A, f ( x), 10, 10 1 2 1 7/ 2 2 4
5
Variational Formulation Find I 2 1 U H0 such that 1 U, V V V : (, ) T 2 B F u v H I 0 where, with, the usual L 2 (Ι) inner product, 2 2 UV B, u, u v, v + a u a v, u a u a v, v V,, F f u g v 11 12 21 22 6
It follows that the bilinear form is coercive, i.e. 2 1 U U U U 2 B H I, E 0 where U 2 2 2 2 2 2 2 2 u v u v E 1, I 1, I 0, I 0, I denotes the energy norm. We also have the a-priori estimate U max 1, A, I 2 2 E 0, I 0, I f g 7
Scale separation: The relationship between and determines the number and nature of the layers. Correspondingly, there are four cases: (I) (II) (III) (IV) The no scale separation case which occurs when neither μ/1 nor ε/μ is small. The 3-scale case in which all scales are separated and occurs when μ/1 is small and ε/μ is small. The first 2-scale case which occurs when μ/1 is not small but ε/μ is small. The second 2-scale case which occurs when μ/1 is small but ε/μ is not small. 8
Theorem 1: There exist constants C, b, δ, q, γ > 0 independent of ε and μ, such that the following assertions are true for the solution U: 9
Theorem 1: There exist constants C, b, δ, q, γ > 0 independent of ε and μ, such that the following assertions are true for the solution U: ( I) U C max n, ( n ) 1/2 n 1, I n 9
Theorem 1: There exist constants C, b, δ, q, γ > 0 independent of ε and μ, such that the following assertions are true for the solution U: ( I) U C max n, ( n ) 1/2 n 1, I n (II) U W U Uˆ R where W C n, R R C e e, I E, I U ( n) n n b/ b / ( n) BL, I BL n ( x) C e BL ˆ ( n) n n dist( x, I )/ BL ( x) C e U n dist( x, I )/ 9
Theorem 1: There exist constants C, b, δ, q, γ > 0 independent of ε and μ, such that the following assertions are true for the solution U: ( I) U C max n, ( n ) 1/2 n 1, I (II) U W U Uˆ R where W C n, R R C e e, I E, I U ( n) n n b/ b / ( n) BL, I BL BL n ( x) C e ˆ ( n) n n dist( x, I )/ BL ( x) C e U n n dist( x, I )/ Additionally, the second component vˆ of sharper estimate Uˆ BL satisfies the 9
2 ( n) n n dist( x, I )/ vˆ ( x) C e 10
(III) 2 ( n) n n dist( x, I )/ vˆ ( x) C e, I E, I If / q, U W Uˆ R where R W b/ ( n) n 1, I Ce BL max n, ˆ ( n) n n dist( x, I )/ BL ( x) C e U R C n 10
(III) 2 ( n) n n dist( x, I )/ vˆ ( x) C e, I E, I If / q, U W Uˆ R where R W b/ ( n) n 1, I R C Ce BL max n, ˆ ( n) n n dist( x, I )/ BL ( x) C e U n Additionally, the second component vˆ of satisfies the sharper estimate Uˆ 2 ( n) n n dist( x, I )/ vˆ ( x) C e BL 10
(IV) U W U R R W U, I E, I ( n) n n ( n) BL, I R BL C n where C( / ) 2 b/ ( x) C / / e e n n dist ( x, I )/ 11
(IV) U W U R R W U, I E, I ( n) n n ( n) BL, I R BL C n where C( / ) 2 b/ ( x) C / / e e n n dist ( x, I )/ Remark: The approximation of U will be constructed according to the above regularity results, i.e. the mesh and polynomial degree distribution will be chosen appropriately, based on the relationship between ε and μ. 11
Discretization As usual, we seek U V 2 H 1 I 2 and we have N N 0 s. t. U, V V V 2 B F V N U U V U V N N 2 V E E N 12
Discretization As usual, we seek U V 2 H 1 I 2 and we have N N 0 s. t. U, V V V 2 B F V N U U V U V N N 2 V E E N The space V N is defined as follows: 12
Discretization As usual, we seek U V 2 H 1 I 2 and we have N N 0 s. t. U, V V V 2 B F V N N U U V U V N 2 V E E N The space V N is defined as follows: Let P n (t) denote the n th Legendre polynomial and define 1 1 1( ) 1, 2( ) 1 2 2 2i 3 1 ( ) P ( t) dt P ( ) P ( ), i 3,..., p 1 i i2 i1 i3 2 1 2(2i 3) 12
Then with Π p the space of polynomials of degree p over [ 1, 1], we have,,,, p span 1 2 3 p 1 13
Then with Π p the space of polynomials of degree p over [ 1, 1], we have,,,, p span 1 2 3 p 1 Now, partition the domain I = (0, 1) by and set 0 1 x0 x1 x M I x, x, h x x, j 1,..., M j j1 j j j j1 13
Then with Π p the space of polynomials of degree p over [ 1, 1], we have,,,, p span 1 2 3 p 1 Now, partition the domain I = (0, 1) by and set 0 1 x0 x1 x M I x, x, h x x, j 1,..., M j j1 j j j j1 Also define the standard (or master) element I ST = ( 1, 1), and note that it can be mapped onto the j th element by the linear mapping 13
1 1 x Q ( ) 1 x 1 x 2 2 j j1 j 14
1 1 x Q ( ) 1 x 1 x 2 2 j j1 j The finite element space V N is then defined as 1 p 0 V, u H I : u Q ( ), j 1,, M N j p j where p p p p,,, M 1 2 degrees assigned to the elements. We have dim VN, p pi 1 is the vector of polynomial M i1 14
An hp finite element method error estimates Definition: Spectral Boundary Layer Mesh For 0, p and 0 1 define 1 2 S, p : VN, p H0 with 0,1, p 1/ 2 0, p, p,1 p,1 p,1, p 1/ 2 0, p,1 p,1, p 1/ 2 p The polynomial degree is taken to be uniformly p over all elements. 15
In practice, the mesh is constructed as follows: If both ε and μ are large (and no layers are present), then [0,1], p If both ε and μ are small (i.e. 0 < ε < μ << 1and both layers are present), then 0, p, p,1 p,1 p,1, p In all other cases, 0, p,1 p,1, p 16
Theorem 2: There exist constants C,, > 0, depending only on the input data, such that U U N Ce p E where U is the exact solution and U N is the finite element solution computed using the Spectral Boundary Layer Mesh. Sketch of Proof: We utilize the previously stated regularity results, along with the following approximation results, the first one being used for the approximation of the smooth part(s) and the boundary layers (within the layer). 17
Lemma 1: Let I j be an interval of length h j and let V C (I ST ) satisfy for some C u, γ u > 0, K 1, V ( n), I j n C max n, K n 1,2,3... u u Then there exist η, β, C > 0, depending only on γ u such hk j that, under the condition, we have p 1/2 hj K h V V V V CC e p 1 j p j p j u 0, I j 0, I where p j j : H 1 (I j ) Π p is a linear operator that satisfies V( I ) V( I ). p j j j n j j p j, 18
The next result is used for the approximation of the boundary layers outside the layer. Lemma 2: Let ν > 0 and let u satisfy u x u x C e x I dist( x, I )/ ( ) ( ) u. Let Δ be an arbitrary mesh on I with mesh points ξ and 1 ξ where ξ (0, ½). Then the piecewise linear interpolant π 1 u Satisfies on (ξ, 1 ξ ): / u 1u u 1u CC 0,(,1 ) ue 0,(,1 ) for some C > 0 independent of ν., 19
The proof is separated in four cases corresponding to the four cases stated in the regularity results. In Case I (asymptotic case), Theorem 1 and Lemma 1 give the desired result. In Case II (3 scale separation), Theorem 1 and Lemma 1 allow us to handle the smooth part and the remainder in the expansion. For the layers, we use Lemma 2 for their approximation outside the layer region and Lemma 1 within. Cases III and IV (2 scale separation) follow with similar combinations of Theorem 1 and Lemmas 1 and 2. 20
Numerical Results We consider the problem with A 2 1 1, 1 2 F 1 We are computing Error 100 U U EXACT FEM EI, U EXACT EI, (An exact solution is available.) 21
We will be comparing the following methods: The h version on a uniform mesh, with p = 1, 2, 3 The p version on a single element, with p = 1, 2, The hp version on the 5 element (variable) mesh, with p = 1, 2, The h version on a Shishkin mesh, with p = 1, 2, 3 The h version on an exponentially graded mesh, with p = 1, 2, 3, where the mesh points are given by 22
with M x j x j x j j0 1 sj x j (2 p 1)ln 1, j 0,, M 2 M 2 s 1 exp (2 p 1) 0 1 23
Percentage Relative Error in the Energy Norm 10 2 10 1 10 0 10-1 10-2 = 0.4, = 1 h-ver, unif mesh, p = 1 h-ver, unif mesh, p = 2 h-ver, unif mesh, p = 3 p-ver, 1 elem slope -1 slope -2 10-3 slope -3 10 0 10 1 10 2 10 3 DOF 24
Percentage Relative Error in the Energy Norm 10 2 10 1 10 0 10-1 10-2 = 0.01, = 0.1 h-ver, unif mesh, p = 1 h-ver, unif mesh, p = 2 h-ver, unif mesh, p = 3 p-ver, 1 elem slope -0.9 slope -1.9 10-3 slope -2.8 10 0 10 1 10 2 10 3 10 4 DOF 25
Percentage Relative Error in the Energy Norm 10 2 10 1 10 0 = 10-7/2 3 10-4, = 0.01 slope -1.5 h-ver, unif mesh, p = 1 h-ver, unif mesh, p = 2 h-ver, unif mesh, p = 3 p-ver, 1 elem slope -0.7 10-1 10 0 10 1 10 2 10 3 10 4 DOF 26
10 2 = 10-7/2 3 10-4, = 0.01 Percentage Relative Error in the Energy Norm 10 1 10 0 10-1 10-2 h-ver, unif mesh, p = 1 p-ver, 1 elem hp-ver, 5 elem h-ver, exp mesh, p = 1 h-ver, Shishkin mesh, p = 1 10 0 10 1 10 2 10 3 DOF slope -1 slope -0.7 slope -0.87 27
10 1 = 10-7/2 3 10-4, = 0.01 Percentage Relative Error in the Energy Norm 10 0 10-1 10-2 10-3 h-ver, unif mesh, p = 1 h-ver, unif mesh, p = 2 h-ver, unif mesh, p = 3 h-ver, Shishkin mesh, p = 1 h-ver, Shishkin mesh, p = 2 h-ver, Shishkin mesh, p = 3 10 0 10 1 10 2 10 3 DOF slope -1 slope -3 slope -2 slope -0.87 slope -1.65 slope -2.22 28
10 1 = 10-7/2 3 10-4, = 0.01 Percentage Relative Error in the Energy Norm 10 0 10-1 10-2 10-3 h-ver, unif mesh, p = 1 h-ver, unif mesh, p = 2 h-ver, unif mesh, p = 3 h-ver, Shishkin mesh, p = 1 h-ver, Shishkin mesh, p = 2 h-ver, Shishkin mesh, p = 3 hp-ver, 5 elem 10 0 10 1 10 2 10 3 DOF 29
10 2 = 10-5, = 10-3 Percentage Relative Error in the Energy Norm 10 1 10 0 10-1 10-2 10-3 h-ver, unif mesh, p = 1 p-ver, 1 elem hp-ver, 5 elem h-ver, exp mesh, p = 1 h-ver, Shishkin mesh, p = 1 10 0 10 1 10 2 10 3 DOF slope -1 slope -0.65 slope -0.9 30
hp-version, 5 elements Percentage Relative Error in the Energy Norm 10 1 10 0 10-1 10-2 3 10-4, = 0.01 = 10-5, = 10-3 = 10-6, = 10-4 10-3 10 1 10 2 DOF 31
Example 2: A 2 1 1 2cos x/ 4 2.2e 2 2 x x 1 x, x 2e 0 f, u(0) u(1) 10x 1 0 (An exact solution is NOT available, hence we use a reference solution.) 32
Percentage Relative Error in the Energy Norm 10 2 10 1 10 0 10-1 = 10-7/2 3 10-4, = 0.01 hp ver, 5 elem h ver, Shishkin mesh, p=1 h ver, exp mesh, p=1 slope -1 slope -0.87 10-2 10 1 10 2 10 3 DOF 33
Percentage Relative Error in the Energy Norm 10 2 = 10-5, = 10-3 10 1 10 0 10-1 hp ver, 5 elem h ver, Shishkin mesh, p=1 h ver, exp mesh, p=1 slope -1 slope -0.87 10-2 10 1 10 2 10 3 DOF 34
Percentage Relative Error in the Energy Norm 10 2 10 1 10 0 10-1 10-2 hp-version, 5 elements 3 10-4, = 0.01 = 10-5, = 10-3 = 10-6, = 10-4 10-3 DOF 10 2 35
Closing Remarks The hp FEM on the Spectral Boundary Layer Μesh yields robust exponential convergence in the energy norm, for the entire range 0 ε μ 1. The regularity theory developed was crucial in the proof of the main approximation result. Extending the current approach to systems with more equations, while conceptually straight forward, appears to be cumbersome. Systems of two equations of convection-diffusion type are currently being investigated. 36