Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Quadratic C 1 -spline collocation for reaction-diffusion problems Torsten Linss 1 Goran Radojev 2 Helena Zarin 2 1 Fakultät für Mathematik und Informatik, FernUniversität in Hagen, Germany 2 Department of Mathematics and Informatics, University of Novi Sad, Serbia "Numerical analysis for Singularly Perturbed Problems" Workshop dedicated to the 60th birthday of Prof. Martin Stynes TU Dresden, November 16-18, 2011
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Outline Introduction (problem, idea) Layer-adapted mesh Interpolation error Collocation method Stability Maximum-norm a priori error bound Maximum-norm a posteriori error bound An adaptive algorithm Numerical experiments Conclusion
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Reaction-diffusion problem Reaction-diffusion problem L u := − ε 2 u ′′ + ru = f in ( 0 , 1 ) , u ( 0 ) = γ 0 , u ( 1 ) = γ 1 (1) ε ∈ ( 0 , 1 ] , r ≥ ̺ 2 > 0 on [ 0 , 1 ]
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Reaction-diffusion problem Reaction-diffusion problem L u := − ε 2 u ′′ + ru = f in ( 0 , 1 ) , u ( 0 ) = γ 0 , u ( 1 ) = γ 1 (1) ε ∈ ( 0 , 1 ] , r ≥ ̺ 2 > 0 on [ 0 , 1 ] 1.5 � � 10 � 2 1.0 � � 10 � 1 0.5 0.2 0.4 0.6 0.8 1.0 1.2
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Idea C. de Boor, B. Swartz ( SIAM J. Numer. Anal , 1973): general theory for spline-collocation methods applied to classical (not SP) BVPs
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Idea C. de Boor, B. Swartz ( SIAM J. Numer. Anal , 1973): general theory for spline-collocation methods applied to classical (not SP) BVPs transition to SPBVPs � bounds with “constants” that tend to infinity when ε → 0 � different approach
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Idea C. de Boor, B. Swartz ( SIAM J. Numer. Anal , 1973): general theory for spline-collocation methods applied to classical (not SP) BVPs transition to SPBVPs � bounds with “constants” that tend to infinity when ε → 0 � different approach quadratic C 1 -splines on a special modified Shishkin mesh
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Idea C. de Boor, B. Swartz ( SIAM J. Numer. Anal , 1973): general theory for spline-collocation methods applied to classical (not SP) BVPs transition to SPBVPs � bounds with “constants” that tend to infinity when ε → 0 � different approach quadratic C 1 -splines on a special modified Shishkin mesh Surla, Uzelac ( ZAMM , 1997): quadratic C 1 -spline collocation, layer-adapted mesh, nodal basis, mesh points as dof’s ⇒ O ( N − 2 ln 2 N )
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Idea C. de Boor, B. Swartz ( SIAM J. Numer. Anal , 1973): general theory for spline-collocation methods applied to classical (not SP) BVPs transition to SPBVPs � bounds with “constants” that tend to infinity when ε → 0 � different approach quadratic C 1 -splines on a special modified Shishkin mesh Surla, Uzelac ( ZAMM , 1997): quadratic C 1 -spline collocation, layer-adapted mesh, nodal basis, mesh points as dof’s ⇒ O ( N − 2 ln 2 N ) LRZ ( submitted to NA , 2011): B-spline basis ⇒ O ( N − 2 ln 2 N )
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Properties of the exact solution The Green’s function �G ξ ( x , · ) � 1 ≤ ( ̺ε ) − 1 , �G ξξ ( x , · ) � 1 ≤ 2 ε − 2 � r G ( x , · ) � 1 ≤ 1 ,
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Properties of the exact solution The Green’s function �G ξ ( x , · ) � 1 ≤ ( ̺ε ) − 1 , �G ξξ ( x , · ) � 1 ≤ 2 ε − 2 � r G ( x , · ) � 1 ≤ 1 , Lemma (Derivative bounds) Let r , f ∈ C 4 [ 0 , 1 ] . Then � 1 + ε − k e − ̺ x /ε + ε − k e − ̺ ( 1 − x ) /ε � � u ( k ) ( x ) � ≤ C � � , for x ∈ ( 0 , 1 ) , k = 0 , . . . , 4. Furthermore, the solution can be decomposed as u = v + w 0 + w 1 . For k = 0 , . . . , 4, the regular � � v ( k ) � solution component v satisfies ∞ ≤ C , while for the layer � parts w 0 and w 1 we have � w ( k ) � w ( k ) � ≤ C ε − k e − ̺ x /ε , � ≤ C ε − k e − ̺ ( 1 − x ) /ε , x ∈ [ 0 , 1 ] . � � � � 0 ( x ) 1 ( x )
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Smoothed Shishkin mesh Shishkin-mesh transition point � σε � λ := min ̺ ln N , q , q ∈ ( 0 , 1 / 2 ) , σ > 0
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Smoothed Shishkin mesh Shishkin-mesh transition point � σε � λ := min ̺ ln N , q , q ∈ ( 0 , 1 / 2 ) , σ > 0 The mesh ∆ : x 0 < x 1 < · · · < x N is generated by x i = ϕ ( i / N ) with the mesh generating function λ q t t ∈ [ 0 , q ] , κ ( t ) := p ( t − q ) 3 + λ ϕ ( t ) := q t t ∈ [ q , 1 / 2 ] , 1 − ϕ ( 1 − t ) t ∈ [ 1 / 2 , 1 ] , where p is chosen such that ϕ ( 1 / 2 ) = 1 / 2. Note, that ϕ ∈ C 1 [ 0 , 1 ] with � ϕ ′ � ∞ , � ϕ ′′ � ∞ ≤ C .
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion The interpolation error for piecewise quadratic splines Notation : There midpoints of the mesh intervals J i := [ x i − 1 , x i ] are denoted with x i − 1 / 2 := ( x i − 1 + x i ) / 2 = x i − 1 + h i / 2 , i = 1 , . . . , N . For, m , ℓ ∈ N , m < ℓ , let � � S m s ∈ C m [ 0 , 1 ] : s | J i ∈ Π ℓ , for i = 1 , . . . , N ℓ (∆) := .
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion S 0 2 -interpolation Given an arbitrary function g ∈ C 0 [ 0 , 1 ] , consider the interpolation problem of finding I 0 2 g ∈ S 0 2 (∆) with I 0 I 0 � � � � 2 g i = g i , i = 0 , . . . , N , and 2 g i − 1 / 2 = g i − 1 / 2 , i = 1 , . . . , N .
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion S 0 2 -interpolation Given an arbitrary function g ∈ C 0 [ 0 , 1 ] , consider the interpolation problem of finding I 0 2 g ∈ S 0 2 (∆) with I 0 I 0 � � � � 2 g i = g i , i = 0 , . . . , N , and 2 g i − 1 / 2 = g i − 1 / 2 , i = 1 , . . . , N . Theorem 1 Assume r , f ∈ C 4 [ 0 , 1 ] . Then the interpolation error u − I 0 2 u for the solution of (1) on a smoothed Shishkin mesh with σ ≥ 3 satisfies CN − 3 ln 3 N , � u − I 0 � � 2 u ∞ ≤ � � � ε 2 max CN − 2 ln 2 N . u − I 0 � ′′ � 2 u � ≤ � � i − 1 / 2 � i = 1 ,..., N
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion S 1 2 -interpolation Given an arbitrary function g ∈ C 0 [ 0 , 1 ] , consider the interpolation problem of finding I 1 2 g ∈ S 1 2 (∆) with I 1 I 1 I 1 � � � � � � 2 g 0 = g 0 , 2 g i − 1 / 2 = g i − 1 / 2 , i = 1 , . . . , N , 2 g N = g N .
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion S 1 2 -interpolation Given an arbitrary function g ∈ C 0 [ 0 , 1 ] , consider the interpolation problem of finding I 1 2 g ∈ S 1 2 (∆) with I 1 I 1 I 1 � � � � � � 2 g 0 = g 0 , 2 g i − 1 / 2 = g i − 1 / 2 , i = 1 , . . . , N , 2 g N = g N . Theorem 2 Assume r , f ∈ C 4 [ 0 , 1 ] . Then the interpolation error u − I 1 2 u for the solution u of (1) on a smoothed Shishkin mesh with σ ≥ 4 satisfies � � ≤ CN − 4 ln 4 N , u − I 1 � � max 2 u � � i � � i = 0 ,..., N ≤ CN − 3 ln 3 N , � u − I 1 � � 2 u � ∞ ε 2 max � � ≤ CN − 2 ln 2 N . u − I 1 � ′′ � 2 u � � i − 1 / 2 � � i = 1 ,..., N
Introduction Layer-adapted mesh The interpolation error The collocation method Numerical experiments Conclusion Let ∆ be an arbitrary partition of [ 0 , 1 ] . Our discretisation is: Find u ∆ ∈ S 1 2 (∆) such that � � u ∆ , 0 = γ 0 , L u ∆ i − 1 / 2 = f i − 1 / 2 , i = 1 , . . . , N , u ∆ , N = γ 1 . Let { ϕ i } N + 1 i = 0 be the B-spline basis in S 1 2 (∆) . Then we may represent u ∆ as N + 1 � u ∆ := α k ϕ k , k = 0 where the α k are determined by collocation.
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