Discrete Sobolev-Poincaré inequalities for Voronoi finite volume approximations - PDF

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W e ie rstra ß -In stitu t fü r A n g e w a n d te A n a ly sis u n d S to c h a stik A. Glitzky (joint work with J. A. Griepentrog) Discrete Sobolev-Poincaré inequalities for Voronoi finite volume approximations

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W e ie rstra ß -In stitu t fü r A n g e w a n d te A n a ly sis u n d S to c h a stik A. Glitzky (joint work with J. A. Griepentrog) Discrete Sobolev-Poincaré inequalities for Voronoi finite volume approximations Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Outline of the talk Notation in finite volume methods Assumptions Potential theoretical lemmas Main result Ideas of the proof of the discrete Sobolev-Poincaré inequality Concluding remarks Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Motivation Sobolev imbedding result u L q c q u H 1 (Ω) u H 1 (Ω) 2n for q [1, ) if n = 2, for q [1, n 2 ] if n 3. Discrete imbedding results in the context of finite volume schemes zero boundary values general boundary values YES NO [1], [2] present talk add. assumpt. d K,σ θd σ, d K,σ θdiam(k) Voronoi finite volume [1] Eymard, Gallouët, Herbin, in Handbook of Numerical Analysis VII [2] Coudière, Gallouët, Herbin, M2AN 35. Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Notation Let Ω R n, n 2, be an open, bounded, polyhedral domain. A Voronoi mesh of Ω denoted by M = (P, T, E) is formed by a family P of grid points in Ω, a family T of Voronoi control volumes, a family E of parts of hyperplanes in R n (surfaces of the V. boxes). For x K P the control volume K of the Voronoi mesh is defined by K = {x Ω : x x K x x L x L P, x L x K }, K T. x K K K σ = K L D m σ L x Kσ σ d L K σ d σ x x K L x L Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Notation The set E and subsets For K, L T with K L either the (n 1) dimensional Lebesgue measure of K L is zero or K L = σ for some σ E. σ = K L denotes the Voronoi surface between K and L. E int denotes the set of interior Voronoi surfaces E ext denotes the set of external Voronoi surfaces For K T : E K is the subset of E such that K = K \ K = σ EK σ. x K K K σ = K L D m σ L x Kσ σ d L K σ d σ x x K L x L Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Notation For σ E: m σ - (n 1)-dimensional measure of the Voronoi surface σ. x σ - center of gravity of σ. d K,σ - Euclidean distance between x K and σ, if σ E K, d σ = x K x L if σ = K L E int. x K K K σ = K L D m σ L x Kσ σ d L K σ d σ x x K L x L half-diamonds D Kσ = {tx K + (1 t)y : t (0, 1), y σ}, mes(d Kσ ) = 1 n m σd K,σ Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Notation Definition. Let M be a Voronoi finite volume mesh of Ω. 1. X(M) = set of functions from Ω to R which are constant on each K T. u K = value of u X(M) on K. 2. Discrete H 1 -seminorm of u X(M) u 2 1,M = where D σ u = u K u L for σ = K L. D σ u 2 m σ, d σ σ E int Aim of the talk: u m Ω (u) L q (Ω) c q u 1,M u X(M), m Ω (u) = 1 mes(ω) Ω u(x) dx. Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Assumptions on the geometry and the mesh (A1) Ω B(0, R) R n open, polyhedral, star shaped w.r.t. some ball B(0, R). {exp { } Let ϱ : R n R2 [0, ), ϱ(y) = R 2 y if y R 2 0 if y R. define ϱ M X(M) as ϱ M K (x) = min ϱ(y) for x K. y K (A2) Let M = (P, T, E) be a Voronoi finite volume mesh with Ω ϱm (x) dx ρ 0 (ρ 0 0) and with the property that E K E ext = x K Ω. (A3) The geometric weights fulfill 0 diam(σ) d σ κ 1 for all σ E int. (A4) There exists a constant κ 2 1 such that max max x K x κ 2 min d K,σ for all x K P. σ E K E int x σ σ E K E int Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Assumptions on the geometry and the mesh (A1) Ω B(0, R) R n open, polyhedral, star shaped w.r.t. some ball B(0, R). {exp { } Let ϱ : R n R2 [0, ), ϱ(y) = R 2 y if y R 2 0 if y R. define ϱ M X(M) as ϱ M K (x) = min ϱ(y) for x K. y K (A2) Let M = (P, T, E) be a Voronoi finite volume mesh with Ω ϱm (x) dx ρ 0 (ρ 0 0) and with the property that E K E ext = x K Ω. (A3) The geometric weights fulfill 0 diam(σ) d σ κ 1 for all σ E int. (A4) There exists a constant κ 2 1 such that max max x K x κ 2 min d K,σ for all x K P. σ E K E int x σ σ E K E int Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Assumptions on the geometry and the mesh (A1) Ω B(0, R) R n open, polyhedral, star shaped w.r.t. some ball B(0, R). {exp { } Let ϱ : R n R2 [0, ), ϱ(y) = R 2 y if y R 2 0 if y R. define ϱ M X(M) as ϱ M K (x) = min ϱ(y) for x K. y K (A2) Let M = (P, T, E) be a Voronoi finite volume mesh with Ω ϱm (x) dx ρ 0 (ρ 0 0) and with the property that E K E ext = x K Ω. (A3) The geometric weights fulfill 0 diam(σ) d σ κ 1 for all σ E int. (A4) There exists a constant κ 2 1 such that max max x K x κ 2 min d K,σ for all x K P. σ E K E int x σ σ E K E int Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Assumptions on the geometry and the mesh (A1) Ω B(0, R) R n open, polyhedral, star shaped w.r.t. some ball B(0, R). {exp { } Let ϱ : R n R2 [0, ), ϱ(y) = R 2 y if y R 2 0 if y R. define ϱ M X(M) as ϱ M K (x) = min ϱ(y) for x K. y K (A2) Let M = (P, T, E) be a Voronoi finite volume mesh with Ω ϱm (x) dx ρ 0 (ρ 0 0) and with the property that E K E ext = x K Ω. (A3) The geometric weights fulfill 0 diam(σ) d σ κ 1 for all σ E int. (A4) There exists a constant κ 2 1 such that max max x K x κ 2 min d K,σ for all x K P. σ E K E int x σ σ E K E int Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Discrete Poincaré inequality Lemma 1. Let Ω R n, n 2, be open, bounded, polyhydral and connected. Then there exists a C 0 0 such that for all Voronoi finite volume meshes M u m Ω (u) L 2 (Ω) C 0 u 1,M u X(M), m Ω (u) = 1 mes(ω) Ω u(x) dx. (Eymard, Gallouët, Herbin, in Handbook of Numerical Analysis VII 2000, Gallouët, Herbin, Vignal, SIAM J. Numer. Anal. 37.) Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Potential theoretical lemmas Lemma 2. Let M be a Voronoi finite volume mesh of Ω such that (A1) (A3) are fulfilled. Let x K0 be a fixed grid point and σ E int an internal Voronoi surface with gravitational center x σ. Then mes({x B(0, R) : [x K0, x] σ }) 1 n diam(ω)n max{2, 4 κ 1 } n 1 m σ x K0 x σ n 1 =: A m σ n x K0 x σ n 1. Idea: Estimation of the solid angle, estimate mes(...) by the measure of the corresponding segment of the ball with radius diam(ω). x K0 x σ σ 0 R Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Potential theoretical lemmas Lemma 3. We assume (A1) (A3). Let { (2, ) if n = 2 q 2n (2, n 2 ) if n 3, 2β = n q n Let x K0 K T P be a fixed grid point. Then σ E K mes(d Kσ) x K0 x σ n 2β max{1+2κ 1, 2} n 2β m n 1 2β (2 R) 2β =: B n n, where m n 1 denotes the measure of the (n 1) dimensional unit sphere in R n. Idea: Show K T σ E K mes(d Kσ) x K0 x σ n 2β c Ω dx ( ). x K0 x n 2β Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Potential theoretical lemmas Lemma 4. We assume (A1) (A4). Let { (2, ) if n = 2 q 2n (2, n 2 ) if n 3, 2β = n q n Let σ E int be a fixed inner Voronoi surface with gravitational center x σ. Then K 0 T σ 0 E K0 mes(d K0σ0) x K0 x σ n qβ ( 1+κ 2 (1+2κ 1 ) ) n qβ m n 1 qβ (2 R) qβ =: D n. Idea: Show K 0 T σ 0 E K0 mes(d K0σ0) x K0 x σ n qβ c Ω dx x x σ n qβ. Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Main result: Discrete Sobolev-Poincaré inequality Theorem 1. Let Ω be an open bounded polyhedral subset of R n and let M be a Voronoi finite volume mesh such that (A1) (A4) are fulfilled. Let q (2, ) for n = 2 2n and q (2, n 2 ) for n 2, respectively. Then there exists a constant c q 0 only depending on n, q, Ω and the constants in (A1) (A4) such that u m Ω (u) L q (Ω) c q u 1,M u X(M), m Ω (u) = 1 mes(ω) Ω u(x) dx. Glitzky, Griepentrog, WIAS-Preprint 1429 (2009) Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Proof of the discrete Sobolev-Poincaré inequality, 1 Let T 0 = {K T : K B(0, R)}. I 1 := (u(x) m Ω (u))ϱ M (x) dx = Ω K T 0 K (u(x) m Ω (u))ϱ M K dx. Let K 0 T be arbitrarily fixed. For all K T 0, f.a.a. x K write u(x) m Ω (u) = u K0 m Ω (u) + (u Ki u Kj )χ σ (x K0, x) σ=k i K j where χ σ (x, y) = { 1 if x, y Ω and [x, y] σ, use correct order! 0 if x / Ω or y / Ω or [x, y] σ =. and [x, y] denotes the line segment {sx + (1 s)y, s [0, 1]}. Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Proof of the discrete Sobolev-Poincaré inequality, 2 Discrete Sobolev s integral representation I 1 = (u K0 m Ω (u)) Ω ϱ M dx+ K T 0 K σ=k i K j (u Ki u Kj )ϱ M K χ σ(x K0, x) dx. By (A2) = I 1 u K0 m Ω (u) I 1 + I 2(K 0 ), ρ 0 ρ 0 I 2 (K 0 ) := K T 0 K (u(x) m Ω (u))ϱ M (x) dx Ω σ=k i K j E int D σ u ϱ M K χ σ(x K0, x) dx. mes(ω) 1/2 u m Ω (u) L 2 (Ω) mes(ω) 1/2 C 0 u 1,M (discrete Poincaré inequality) Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Proof of the discrete Sobolev-Poincaré inequality, 3 I 2 (K 0 ) = D σ u ϱ M K χ σ(x K0, x) dx σ E int K T K 0 D σ u mes({x B(0, R) : σ [x K0, x] }) σ E int m σ A n D σ u x K0 x σ n 1 Lemma 2 σ E int x K0 σ Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Proof of the discrete Sobolev-Poincaré inequality, 4 Hölder s inequality for α 1 = q, α 2 = 2q/(q 2), α 3 = 2, let 2β = n q n I 2 (K 0 ) A n D σ u x K0 x σ 1 n m σ σ E int ( D σ u 2 x K0 x σ n+qβ m ) σ 1/q( D σ u 2 m ) q 2 σ 2q d σ d σ σ E int σ E int ( ) 1/2 x K0 x σ n+2β m σ d K,σ K T σ E K ( Bn 1/2 u 1 2/q 1,M σ E int D σ u 2 x K0 x σ n+qβ m σ d σ ) 1/q Lemma 3, discrete H 1 -seminorm Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Proof of the discrete Sobolev-Poincaré inequality, 5 I 2 q L q (Ω) = K 0 T A q nbn q/2 u q 2 1,M I 2 (K 0 ) q mes(d K0 σ 0 ) σ 0 E K0 D σ u 2 m σ d σ σ E int K 0 T σ 0 E K0 x K0 x σ n+qβ mes(d K0 σ 0 ) A q nb q/2 n D n u q 1,M Lemma 4, discrete H 1 -seminorm In summary, for u X(M) u m Ω (u) L q (Ω) 1 ρ 0 [ I 1 L q (Ω) + I 2 L q (Ω) 1 ρ 0 mes(ω) 1/q+1/2 C 0 u 1,M + A n ρ 0 B 1/2 n D 1/q n u 1,M ] Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Concluding remarks For q [1, 2] and n 2, the discrete Sobolev-Poincaré inequalities u m Ω (u) L q (Ω) c q u 1,M u X(M) are a direct consequence of Theorem 1 and Hölder s inequality. 2n Corollary. Assume (A1) (A4). Let q [1, ) for n = 2 and q [1, n 2 ) for n 3, respectively. Then there exists a constant c q 0 only depending on n, q, Ω and the constants in (A1) (A4) such that u L q (Ω) c q u 1,M + mes(ω) 1 1 q u dx u X(M). More general domains: Discrete Sobolev inequalities remain true if Ω is a finite union of δ-overlapping star shaped domains Ω i, i = 1,..., r. Ω Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Concluding remarks For q [1, 2] and n 2, the discrete Sobolev-Poincaré inequalities u m Ω (u) L q (Ω) c q u 1,M u X(M) are a direct consequence of Theorem 1 and Hölder s inequality. 2n Corollary. Assume (A1) (A4). Let q [1, ) for n = 2 and q [1, n 2 ) for n 3, respectively. Then there exists a constant c q 0 only depending on n, q, Ω and the constants in (A1) (A4) such that u L q (Ω) c q u 1,M + mes(ω) 1 1 q u dx u X(M). More general domains: Discrete Sobolev inequalities remain true if Ω is a finite union of δ-overlapping star shaped domains Ω i, i = 1,..., r. Ω Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Concluding remarks For q [1, 2] and n 2, the discrete Sobolev-Poincaré inequalities u m Ω (u) L q (Ω) c q u 1,M u X(M) are a direct consequence of Theorem 1 and Hölder s inequality. 2n Corollary. Assume (A1) (A4). Let q [1, ) for n = 2 and q [1, n 2 ) for n 3, respectively. Then there exists a constant c q 0 only depending on n, q, Ω and the constants in (A1) (A4) such that u L q (Ω) c q u 1,M + mes(ω) 1 1 q u dx u X(M). More general domains: Discrete Sobolev inequalities remain true if Ω is a finite union of δ-overlapping star shaped domains Ω i, i = 1,..., r. Ω Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Concluding remarks Critical exponent in higher space dimensions: For n 3, the discrete version of the Sobolev imbedding H 1 (Ω) L 2n n 2 (Ω) (the critical Sobolev exponent in n dimensions) can not be obtained by the presented technique using the Sobolev integral representation. This is exactly the same situation as for the continuous case. Discrete Sobolev-Poincaré inequalities for p n, q [1, u m Ω (u) L q (Ω) c q u 1,p,M np n p ): u X(M) - requires a Poincaré like inequality using the discrete W 1,p -seminorm u 1,p,M = ( σ E int ( Dσ u d σ ) p mσ d σ ) 1/p. - adapt technique of the proof of Theorem 1 for p 2. Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20) Concluding remarks Critical exponent in higher space dimensions: For n 3, the discrete version of the Sobolev imbedding H 1 (Ω) L 2n n 2 (Ω) (the critical Sobolev exponent in n dimensions) can not be obtained by the presented technique using the Sobolev integral representation. This is exactly the same situation as for the continuous case. Discrete Sobolev-Poincaré inequalities for p n, q [1, u m Ω (u) L q (Ω) c q u 1,p,M np n p ): u X(M) - requires a Poincaré like inequality using the discrete W 1,p -seminorm u 1,p,M = ( σ E int ( Dσ u d σ ) p mσ d σ ) 1/p. - adapt technique of the proof of Theorem 1 for p 2. Discrete Sobolev-Poincaré inequalities ÖMG-DMV Congress, September 21, (20)
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