A posteriori error estimates for non-linear parabolic equations

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1 A posteriori error estimates for non-linear parabolic equations R Verfürt Fakultät für Matematik, Rur-Universität Bocum, D-4478 Bocum, Germany address: rv@num1rur-uni-bocumde Date: December 24 Summary: We consider space-time discretizations of non-linear parabolic equations Te temporal discretizations in particular cover te implicit Euler sceme and te mid-point rule For linear equations tey correspond to te well-known A-stable θ-scemes Te spatial discretizations consist of standard conforming finite element spaces tat can vary from one time-level to te oter Te spatial meses may be locally refined, but must be isotropic For tese discretizations we derive a residual a posteriori error estimator wic yields upper and lower bounds on te error Te ratio of upper and lower bounds does not depend on any mes-size in space or time nor on any relation between bot In particular tere is no restriction on te relative size of te temporal and spatial mes-sizes Key words: a posteriori error estimates; non-linear parabolic equations; space-time finite elements AMS Subject classification: 65N3, 65N15, 65J15 1 Introduction We consider non-linear parabolic equations u div a(x, u, u) + b(x, u, u) = in Ω (, T ] u = on Γ (, T ] u(, ) = u in Ω (11) in a bounded space-time cylinder wit a convex two-dimensional polygonal crosssection Ω IR 2 aving a Lipscitz boundary Γ Te final time T is arbitrary, but kept fixed in wat follows Te coefficients a : Ω IR IR 2 IR 2 and b : Ω IR IR 2 IR must be continuously differentiable wit Lipscitz-continuous derivatives Tey ave to satisfy suitable growt conditions so tat problem (11) admits an appropriate variational formulation (cf Sections 2 and 4 for details) Some sample problems satisfying tese conditions are given in Section 4 We consider space-time discretizations of problem (11) Te temporal discretizations in particular cover te implicit Euler sceme and te mid-point rule For linear problems tey correspond to te well-known A-stable θ-scemes Te spatial discretizations consist of standard conforming finite element spaces tat can vary from 1

2 one time-level to te oter Te spatial meses may be locally refined, but must be isotropic For tese discretizations we derive a residual a posteriori error estimator It consists of two contributions: a spatial error estimator and a temporal one Te spatial contribution is a standard residual a posteriori error estimator for te non-linear elliptic equation arising from te time-discretization of problem (11) It consists of an element residual on a space-time cylinder K [t n 1, t n ] and of an edge residual on te lateral boundary K [t n 1, t n ] Te evaluation of te temporal error estimator requires at eac time-level te solution of a discrete Poisson problem Tis term can be interpreted as an edge residual on te bottom K t n 1 } measured in an appropriate dual norm Te additional work of solving a supplementary discrete problem at eac time-level is te price we ave to pay for making computable tis dual norm Tis extra work is comparable to te one required by te now popular estimators tat are based on te solution of suitable discrete adjoint problems [3] We prove tat te error estimator yields upper and lower bounds for te error measured in a suitable L r (, T ; W 1,ρ (Ω))-norm (cf Section 2 for a definition of tese spaces and teir norms) Te ratio of te upper and lower bounds does not depend on any mes-size in space or time nor on any relation between tese parameters Te present results sould be compared to our old results in [1]: (1) Here, we consider standard time-discretizations wic in particular cover te implicit Euler sceme and te midpoint-rule In [1], we used non-standard timediscretizations wic could be interpreted as implicit Runge-Kutta scemes and wic covered te Crank-Nicolson sceme as metod of lowest order (2) Here, te ratio of upper and lower error bounds is independent of any mes-size in space and time and of any relation between bot parameters In [1], te ratio of te upper and lower error bounds is proportional to τ τ were and τ denote te local mes-sizes in space and time respectively (3) Te present analysis and te one in [1] bot depart from an abstract non-linear equation F (u) = wit a continuously differentiable mapping F : X Y between appropriate function spaces (Y denoting te dual of Y ) Here, X carries a stronger topology tan Y (cf Section 2), in [1] tese rôles are reversed (4) In [1] we do not ave to solve additional discrete problems in order to evaluate te error estimator Te article is organized as follows In Section 2 we introduce te relevant function spaces and teir norms Section 3 gives te finite element discretization Departing from te abstract error estimate of [9, Proposition 21] we prove in Section 4 tat te error is equivalent to a residual wic is defined in a suitable dual space Tis residual is split into two contributions: a spatial residual and a temporal one In Section 5 2

3 we derive upper and lower bounds for te spatial residual Te temporal residual is treated in Section 6 Combining tese results we obtain in Section 7 a preliminary a posteriori error estimator It yields upper and lower bounds on te error wic are independent of te mes-sizes in te sense described above Tis preliminary error estimator, owever, is not suited for practical computations since it incorporates te dual norm of a suitable residual Te residual itself is easy to evaluate, but te dual norm is not directly accessible To overcome tis difficulty we present in Section 8 some W 1,q -stability results for te Laplacian bot in analytic and discrete form Tese results require tat te cross-section Ω is two-dimensional and convex Based on tese results we present in Section 9 te error estimator in its final form Te computation of te dual norm ere is replaced by te evaluation of te solution of an auxiliary discrete Poisson problem 2 Function spaces For any bounded open subset ω of Ω wit Lipscitz boundary γ we denote by W k,p (ω), k IN, 1 p, L p (ω) = W,p (ω), and L p (γ) te usual Sobolev and Lebesgue spaces equipped wit te standard norms (cf [1], [6, Vol 3, Cap IV ]): u k,p;ω = u k, ;ω α k ω D α u(x) p dx = max esssup D α u(x) α k x ω 1,p, p <, and u p;γ = u ;γ γ u(x) p ds(x)} 1,p, p <, = esssup u(x) x γ Here, α IN 2 is a multi-index, α = α 1 + α 2, and ds denotes te lengt element of te curve γ Set W 1,p (Ω) = u W 1,p (Ω) : u = on Γ } and denote its dual space by W 1,p (Ω) = W 1,p (Ω) for 1 < p < Here and in te sequel, p denotes te dual exponent of p defined by 1 p + 1 p = 1 Te duality pairing between W 1,p (Ω) and W 1,p (Ω) will always be denoted by, were te relevant Lebesgue exponent p will be apparent from te context 3

4 Let V and W be two Banac spaces suc tat V is continuously embedded in W Given two real numbers a and b wit a < b, we denote by L p (a, b; V ), 1 p, te space of all measurable functions u defined on (a, b) wit values in V suc tat te mapping t u(, t) V is in L p ((a, b)) L p (a, b; V ) is a Banac space equipped wit te norm b u Lp (a,b;v ) = u(, t) p V dt, p <, u L (a,b;v ) a = esssup u(, t) V t (a,b) (cf [6, Vol 5, Cap XVIII, 1]) Sligtly canging te notation of [6], we furter introduce te Banac space W p (a, b; V, W ) = equipped wit te norm u L p (a, b; V ) : u } Lp (a, b; W ) b b u W p (a,b;v,w ) = u(, t) p V dt + u W (a,b;v,w ) a = esssup t (a,b) a u (, t) p W dt, p < max u(, t) V, u (, t) W Here te partial derivative u must be interpreted in te distributional sense [6, loccit] If p > 1, we know from [6, Vol 5, Cap XVIII, 1, Proposition 9] tat for any u W p (a, b; V, W ) te traces u(, a) and u(, b) are defined as elements of W A function u is called a weak solution of problem (11) if tere are parameters r, p, ρ, π (1, ) suc tat u W r (, T ; W 1,ρ (Ω), W 1,π (Ω)), } u(, ) = u in W 1,π (Ω) (21) and T u T (, t), v(, t) dt + a(x, u, u) v + b(x, u, u)v} dxdt = Ω v L p (, T ; W 1,π (Ω)) (22) (cf [2]) Note tat W 1,ρ (Ω) is continuously embedded in W 1,π (Ω) for all ρ and π since Ω IR 2 4

5 3 Finite element discretization For te discretization we coose an integer N 1 and intermediate times = t < t 1 < < t N = T and set τ n = t n t n 1, 1 n N Wit eac intermediate time t n, n N, we associate a partition T,n of Ω and a corresponding finite element space X,n Tese ave to satisfy te following conditions: (1) Affine equivalence: every element K T,n is eiter a triangle or a parallelogram (2) Admissibility: any two elements are eiter disjoint or sare a vertex or a complete edge (3) Sape regularity: for any element K te ratio of its diameter K to te diameter ρ K of te largest inscribed ball is bounded uniformly wit respect to all partitions T,n and to N (4) Transition condition: for 1 n N tere is an affinely equivalent, admissible, and sape-regular partition T,n suc tat it is a refinement of bot T,n and T,n 1 and suc tat sup 1 n N sup K T,n sup K T,n ;K K K K < (5) Eac X,n consists of continuous functions wic vanis on Γ and wic are piecewise polynomials, te degrees being bounded uniformly wit respect to all partitions T,n and to N (6) Eac X,n contains te space of continuous, piecewise linear finite elements corresponding to T,n Triangular and quadrilateral elements may be mixed Condition (2) excludes anging nodes Condition (3) is a standard one and allows for locally refined meses However, it excludes anisotropic elements Condition (4) is due to te simultaneous presence of finite element functions defined on different grids Usually te partition T,n is obtained from T,n 1 by a combination of refinement and of coarsening In tis case Condition (4) only restricts te coarsening It must not be too abrupt nor too strong We coose a parameter θ [ 1 2, 1] and keep it fixed in wat follows Ten te space-time discretization of problem (11) consists in finding u n X,n, n N, suc tat u = π u (31) and, for n = 1,, N, and all v X,n Ω u n un 1 v dx + a(x, u nθ, u nθ ) v + b(x, u nθ, u nθ } )v dx = (32) τ n Ω were u nθ = θu n + (1 θ)u n 1 (33) 5

6 and π denotes te L 2 -projection onto X, Wit every solution (u n ) n N of problems (31) and (32) we associate two functions u τ and ũ τ Te function u τ is piecewise affine on te intervals [t n 1, t n ], 1 n N, and equals u n at time t n Te function ũ τ is piecewise constant on te intervals (t n 1, t n ], 1 n N, and equals u nθ on (t n 1, t n ] Since te function t u τ (, t) is continuous and piecewise affine wit values in W 1,ρ (Ω), it is differentiable in te distributional sense [6, Vol 5, Cap XVIII, 1] and its weak derivative satisfies u τ = un un 1 on (t n 1, t n ) (34) τ n 4 Te equivalence of error and residual As usual for non-linear problems, our a posteriori error estimates are based on te abstract error estimate of [9, Proposition 21] For completeness we sortly recall tis result Given two Banac spaces X and Y wit norms X and Y we denote by L(X, Y ) te space of continuous linear mappings of X into Y and equip it wit its standard norm Ax Y A L(X,Y ) = sup x X\} x X ISOM(X, Y ) denotes te space of all invertible A L(X, Y ) wit A 1 L(Y, X) Given a continuously differentiable map F : X Y of X into te dual Y of Y we look for solutions of te non-linear equation F (u) = (41) 41 Lemma [9, Proposition 21] Let u X be a solution of problem (41) Assume tat u is regular, ie DF (u) ISOM(X, Y ) and tat DF is locally Lipscitz continuous at u, ie tere are constants γ > and R > suc tat DF (v) DF (w) L(X,Y ) γ v w X olds for all v, w X wit v u X R and w u X R Set } R = min R, γ 1 DF (u) 1 1 L(Y,X), 2γ 1 DF (u) L(X,Y ) Ten te following error estimate olds for all v X wit v u R: 1 DF (u) 1 2 L(X,Y ) F (v) Y v u X 2 DF (u) 1 L(Y,X) F (v) Y 6

7 For abbreviation we define a function G : L r (, T ; W 1,ρ (Ω)) L p (, T ; W 1,π (Ω)) by G(u), v = a(x, u, u) v + b(x, u, u)v} dt ae in (, T ) (42) Ω Ten te weak formulation (21), (22) of problem (11) fits into te framework of Lemma 41 wit We terefore obtain: X = W r (, T ; W 1,ρ (Ω), W 1,π (Ω)), Y = W 1,π (Ω) L p (, T ; W 1,π (Ω)), ( ) u(, ) u, v 1 F (u), (v 1, v 2 ) = u, v 2 + G(u), v 2 } dt T (43) 42 Lemma Let u and u τ be solutions of problems (21), (22) and (31), (32) Assume tat u and te function F of equation (43) satisfy te conditions of Lemma 41 and tat u u uτ W r (,T ;W 1,ρ (Ω),W 1,π (Ω)) R Ten tere are two constants c and c suc tat c u u 1,π + u } τ + G(u τ ) L p (,T ;W 1,π (Ω)) u u τ W r (,T ;W 1,ρ (Ω),W 1,π (Ω)) c u u 1,π + u τ + G(u τ ) Lp (,T ;W 1,π (Ω)) 43 Remark Assume tat G is locally Lipscitz continuous at te solution u of problems (21), (22), ie, tere are two constants γ > and R > suc tat } DG(v) DG(w) L(Lr (,T ;W 1,ρ (Ω)),L p (,T ;W 1,π (Ω))) γ v w L r (,T ;W 1,ρ (Ω)) olds for all v, w L r (, T ; W 1,ρ (Ω)) wit u v Lr (,T ;W 1,ρ (Ω)) R and u w L r (,T ;W 1,ρ (Ω)) R Ten te function F of equation (43) satisfies te Lipscitz condition of Lemma 41 wit te same constants γ and R Some examples of problems following into te present category are given by: (1) Te eat equation wit convection and non-linear diffusion coefficient: a(x, u, u) = k(u) u, b(x, u, u) = c u f, f L (Ω), c C(Ω, IR 2 ), k C 2 (IR), k(s) α >, k (l) (s) γ s IR, l, 1, 2}, ρ = π (2, 4), p > 2, r 2p 7

8 (2) Te non-stationary equation of prescribed mean curvature: a(x, u, u) = [1 + u 2 ] 1/2 u, b(x, u, u) = f L 2 (Ω), ρ > 2, π = ρ 2, r 2ρ, p = r 2 (3) Te non-stationary α-laplacian: a(x, u, u) = u α 2 u, α 2, b(x, u, u) = f L α (Ω), ρ = α, π = α, r > 6, p = r 3 (4) Te non-stationary subsonic flow of an irrational, ideal, compressible gas: [ a(x, u, u) = 1 γ 1 ] 1/(γ 1) u 2 u, γ > 1, 2 b(x, u, u) = f L π (Ω), ρ = 2γ γ 1, π = 2γ γ + 1, r > 6, p = r 3 Lemma 42 states tat te error u u τ and te residual u τ +G(u τ ) are equivalent In te following sections we will derive computable upper and lower bounds on te residual To tis end we split it into a spatial and a temporal contribution and set R (u τ ), v = u τ + G(ũ τ ), v (44) and R τ (u τ ), v = G(ũ τ ) G(u τ ), v (45) Obviously we ave u τ + G(u τ ) = R (u τ ) + R τ (u τ ) (46) Since u τ (32) is equivalent to = un un 1 τ n (cf (34)) and ũ τ = u nθ on eac interval (t n 1, t n ], problem R (u τ ), v = v X n, 1 n N (47) 8

9 5 Estimation of te spatial residual For te estimation of te spatial residual R (u τ ) we need some additional notations We denote by Ẽ,n, 1 n N, te set of all edges of T,n Wit eac edge E Ẽ,n we associate a unit vector n E ortogonal to E suc tat it points to te outward of Ω if E is part of te boundary For every edge E tat is not contained in te boundary we denote by [] E te jump across E in direction n E Te quantity [] E of course depends on te orientation of n E, but quantities of te form [n E ] E are independent tereof Wit eac edge E tat is not contained in te boundary, we associate te set ω E wic is te union of te two elements tat sare E If E is a boundary-edge, ω E simply is te unique element tat as E as an edge For every n between 1 and N we denote by N,n te set of all element vertices in T,n tat do not lie on te boundary Wit every vertex x N,n we associate te nodal bases function λ x wic is uniquely defined by te properties λ x K R 1 (K) K T,n, λ x (y) = y N,n \x}, λ x (x) = 1 Here, as usual, R k (K) denotes te set of all polynomials of total degree k, if K is a triangle, and of maximal degree k, if K is a quadrilateral Te support of a nodal bases function λ x is denoted by ω x and consists of all elements in T,n tat sare te vertex x Denote by π x te L 2 (ω x )-projection onto R 1 defined by π x vw = vw w R 1 ω x ω x Ten te interpolation operator I,n of Clément [5] corresponding to T,n is defined by I,n v = λ x (π x v)(x) (51) x N,n Due to condition (6) of Section 3 I,n maps L 1 (Ω) into a subspace of X,n 51 Lemma [9, Lemma 31] Te following error estimates old for all 1 p, all v W 1,p (Ω), all 1 n N, all K T,n, and all E E,n : v I,n v,p;k c 1 K v 1,p; ω k, v I,n v p;e c p E v 1,p; ω E Here ω K and ω E consists of all elements in T,n tat sare at least a vertex wit K or E, respectively Te constants c 1 and c 2 only depend on te ratios K /ρ K For every element K T,n and every edge E Ẽ,n we denote by N K and N E te set of its vertices and set ψ K = γ K x N K λ x, ψ E = γ E x N E λ x 9

10 Te constants γ k and γ E are cosen suc tat ψ K and ψ E equal 1 at te barycentre of K and E, respectively Te support of ψ K is contained in K and ψ K, ;K = 1 Similarly, te support of ψ E is contained in ω E and ψ E, ;ωe = ψ E ;E = 1 52 Lemma [9, Lemma 33] Te following estimates old for all 1 p, all k IN, all 1 n N, all K T,n, all v R k (K), all E Ẽ,n, and all σ R k (E): K vψ Kw c 3 v,p;k, w,p ;K sup w R k (K) sup η R k (E) ψ K v 1,p;K E σψ Eη η p ;E c 4 1 K v,p;k, c 5 σ p;e, ψ E σ E 1,p;ωE c p E σ p;e, ψ E σ,p;ωe c 7 1/p E σ p;e Here, a polynomial σ defined on an edge is continued in te canonical way to a polynomial defined on IR 2 Te constants c 3,, c 7 only depend on te polynomial degree k and on te ratios K /ρ K We coose an integer l and denote for every n between 1 and N by a,n (x, u nθ u nθ ) and b,n(x, u nθ, unθ ) te L2 -projections of a(x, u nθ, unθ ) and b(x, unθ ) onto discontinuous vector-fields respectively functions wic are piecewise u nθ polynomials of degree l on te elements of T,n Wit tis notation we define element residuals R K, K T,n, 1 n N, by R K = un un 1 τ n,, div a,n (x, u nθ, u nθ ) + b,n (x, u nθ, u nθ ), (52) edge residuals R E, E Ẽ,n, 1 n N, by [ne a R E =,n (x, u nθ, unθ )] E if E Γ, if E Γ, (53) elementwise data errors D K, K T,n, 1 n N, by D K =a(x, u nθ, u nθ ) a,n (x, u nθ, u nθ ) + b(x, u nθ, u nθ ) b,n (x, u nθ, u nθ ), and edgewise data errors D E, E Ẽ,n, 1 n N, by (54) D E = [ne (a(x, u nθ, unθ ) a,n(x, u nθ, unθ ))] E if E Γ, if E Γ (55) Te coice of te parameter l is influenced by te polynomial degree of te finite element spaces X,n and by te smootness of te coefficients a, b Te simplest coice of course is l = Wit tese preparations we are now ready to bound te spatial residual 1

11 53 Lemma For every n between 1 and N define a spatial error indicator η n by η n = π K R K π,π;k + E R E π π;e K T,n E Ẽ,n 1/π (56) and a spatial data error indicator Θ n by Θ n = π K D K π,π;k + E D E π π;e K T,n E Ẽ,n 1/π (57) Ten tere are functions w n W 1,π (Ω), 1 n N, and constants c and c suc tat on eac interval (t n 1, t n ], 1 n N, te following estimates old: R (u τ ) 1,π c η n + Θ n } (58) and (η n ) π R (u τ ), w n + Θ n w n 1,π, w n 1,π c (η n ) π 1 (59) Te constants c and c depend on te ratios K /ρ K Te constant c in addition depends on te ratios K / K in condition (4) of Section 3 Te constant c in addition depends on te polynomial degree l Proof Coose an integer n between 1 and N and keep it fixed in wat follows Integration by parts on te elements of T,n yields te following L 2 -representation of te residual R (u τ ), v = R K v + R E v K T,n + K T,n K K D K v + E Ẽ,n E E Ẽ,n E D E v Lemma 51 and Hölder s inequality terefore imply for all v W 1,π (Ω) (51) R (u τ ), v I,n v c v 1,π η n + Θ n } (511) Te constant c only depends on te constants c 1 and c 2 of Lemma 51 and te ratios K /ρ K Since I,n maps L 1 (Ω) into a subspace of X,n, equations (47) and (511) prove te upper bound (58) 11

12 From Lemma 52 we conclude tat for every element K T,n E Ẽ,n tere are polynomials v K and σ E suc tat and every edge K R K ψ K v K v K,π ;K R E ψ E σ E E σ E π ;E = R K π,π;k, c 1 3 R K π 1,π;K, = R E π π;e, c 1 5 R E π 1 π;e Set w n = γ 1 π Kψ K v K + γ 2 K T,n E Ẽ,n E ψ E σ E Te constants γ 1 and γ 2 are arbitrary at present and will be determined below Te subsequent arguments are based on te following observations: te supports of te ψ K are mutually disjoint, te support of a ψ K intersects te support of at most four different ψ E s, te support of a ψ E intersects te support of at most two ψ K s, te support of a ψ E intersects te support of almost two oter ψ E s Since (π 1)π = π, Lemma 52 terefore yields Tis proves tat w n π 1,π = w n π K T,n 5 π 1 γ π π 1 γ π 2 5 π 1 γ π π 1 γ π 2 1,π ;K K T,n ππ K ψ K v K π 1,π ;K K T E K,n c π 3 c π 4 (π 1)π K K T,n K T E K,n π E ψ E σ E π 1,π ;K c π 5 c π R K π (π 1),π;K 6 π +1 π E 5 π maxγ 1, γ 2 } π maxc 1 3 c 4, c 1 5 c 6} π (η n ) π R E π (π 1) π;e = 5 π maxγ 1, γ 2 } π maxc 1 3 c 4, c 1 5 c 6} π (η n ) (π 1)π w n 1,π 5 maxγ 1, γ 2 } maxc 1 3 c 4, c 1 5 c 6}(η n ) π 1 (512) 12

13 Since E K for all edges E of any element K, Lemma 52 also implies tat K T,n = γ 1 γ 1 K K T,n π K + γ 2 R K w n + E Ẽ,n E + γ 2 K E Ẽ,n R K ψ K v K E K T E K,n R E ψ E σ E E K T,n π K R K π,π;k + γ 2 K E R E w n R K ψ E σ E E R E π π;e E Ẽ,n γ 2 c 1 5 c 7 K 1 π E R K,π;K R E π 1 π;e K T E K,n Using Young s inequality ab 1 π aπ + 1 π b π wit a = ε 1 π c 1 5 c 7 K R K,π;K and b = ε 1 π 1 π E R E π 1 π;e and arbitrary ε > and taking into account tat π (π 1) = π, tis gives γ 1 K T,n K R K w n + E Ẽ,n K T,n π K R K π,π;k + γ 2 E R E w n E R E π π;e E Ẽ,n 1 γ 2 π ε π π c π 5 c π 7 π K R K π,π;k + ε } π E R E π π;e K T E K,n (γ 1 4 γ 2 π ε π π c π 5 cπ 7 ) + γ 2 (1 2 ε π ) K T,n π K R K π,π;k E Ẽ,n E R E π π;e 13

14 Tis proves tat K T,n K R K w n + E Ẽ,n E R E w n 4 min γ 1 γ 2 π ε1 π c π 5 cπ 7, γ 2 (1 2ε } π ) (η) n π (513) From Lemma 52 we also obtain K T,n = γ 1 γ 1 K K T,n π K + γ 2 D K w n + K K T E K,n E Ẽ,n E D K ψ K v K + γ 2 E K D E w n D K ψ E σ E E Ẽ,n E K T,n c 1 3 π K D K,π;K R K π 1,π;K + γ 2 E Ẽ,n c 1 + γ 2 K T,n c 1 5 E D E π;e R E π 1 π;e 5 c 7 K 1 π E D K,π;K R E π 1 π;e E D E ψ E σ E Applying Hölder s inequality and using once more te relation π (π 1) = π, tis proves D K w n + D E w n K E (514) K T,n E Ẽ,n maxγ 1, γ 2 } maxc 1 3, c 1 5, c 1 5 c 7}5Θ n (η n ) π 1 Now we coose (recall tat 1 < π, π < ) ε = 1 2, γ 2 = π π 1, γ 1 = π 1 c π 5 cπ 7 Tis gives 4 min γ 1 γ 2 π ε1 π c π 5 cπ 7, γ 2 (1 2ε } π ) = 1 (515) Equations (51), (512), (513), (514), and (515) prove estimate (59) 14

15 6 Estimation of te temporal residual Recall tat te function u τ is piecewise affine on te intervals [t n 1, t n ] and equals u n at time t n and tat te function ũ τ is piecewise constant on te intervals (t n 1, t n ] and equals u nθ = θun + (1 θ)un 1 on (t n 1, t n ] Te following lemma provides us wit sarp upper and lower bounds for te temporal residual 61 Lemma Define te residual rτ n W 1,π (Ω), 1 n N, by rτ n, v = v a p (x, u nθ, nθ ) (u n u n 1 ) Ω + v a u (x, u nθ, u nθ )(u n u n 1 ) + vb p (x, u nθ, u nθ ) (u n u n 1 + vb u (x, u nθ, u nθ )(u n u n 1 ) ) } dx (61) v W 1,π (Ω), were te indices u and p denote te derivatives of te corresponding function wit respect to te second respectively tird variable Assume tat te function G defined in equation (42) satisfies te Lipscitz condition of Remark 43 and tat u u τ Lr (,T ;W 1,ρ (Ω)) R and u ũ τ Lr (,T ;W 1,ρ (Ω)) R Ten te following upper and lower bounds for te temporal residual are valid: R τ (u τ ) Lp (,T ;W 1,π (Ω)) and N } 1,p τ n rτ n p 1,π + γ 2 N τ n rτ n p 1,τ N τ n u n u n 1 r 1,ρ } 2/r (62) 2p + 1 R τ (u τ ) Lp (,T ;W 1,π (Ω)) (63) + γ 2 N τ n u n u n 1 r 1,ρ } 2/r } Proof From te definition (45) of te temporal residual we immediately obtain R τ (u τ ) = G(ũ τ ) G(u τ ) = 1 DG(u τ + s(ũ τ u τ )(ũ τ u τ )ds = DG(ũ τ )(ũ τ u τ ) + 1 [DG(u τ + s(ũ τ u τ )) DG(ũ τ )](ũ τ u τ )ds = R (1) τ (u τ ) + R (2) τ (u τ ) 15 (64)

16 From te definition of te functions u τ and ũ τ we conclude tat ũ τ u τ = (θ t t n 1 )(u n u n 1 ) on (t n 1, t n ] (65) τ n A straigt forward calculation yields for all q (1, ) and all n between 1 and N tn θ t t n 1 t n 1 τ n q dt = τ n 1 Equations (65) and (66) in particular imply u τ ũ τ L r (,T ;W 1,ρ (Ω)) θ z q dz = τ n 1 q + 1 θq+1 + (1 θ) q+1 } (66) N τ n u n u n 1 r 1,ρ} 1/r Tis estimate and te Lipscitz continuity of DG yield an upper bound for te term R (2) τ in equation (64): R (2) τ (u τ ) Lp (,T ;W 1,π (Ω)) 1 2 γ u τ ũ τ 2 L r (,T ;W 1,ρ (Ω)) 1 2 γ N τ n u n u n 1 r 1,ρ} 2/r (67) Since tis is te second term on te rigt-and sides of estimates (62) and (63), it remains to prove tat R τ (1) (u τ ) Lp (,T ;W 1,π (Ω)) is bounded from above and from below by te corresponding multiples of N τ n rτ n p 1,π }1/p Equations (42), (61), and (65) yield R τ (1) (u τ ) = (θ t t n 1 )rτ n on (t n 1, t n ] (68) τ n Combining tis wit equation (66) and observing tat 2 q θ q+1 + (1 θ) q+1 1 for all q (1, ) and all θ [ 1 2, 1], we obtain te estimate } 1/p 1 N p p τ n r n τ p 1,π R τ (1) (u τ ) L p (,T ;W 1,π (Ω)) N 1/p τ n rτ n 1,π} p Tis completes te proof of estimates (62) and (63) 16

17 7 A preliminary a posteriori error estimate Te following lemma gives a posteriori error bounds wic are reliable and efficient in te sense described in te Introduction However, tey are not suited for practical computations since tey incorporate Sobolev norms of a negative order In Section 9 we will bound tese terms by computable quantities 71 Lemma Assume tat te functions F, G, u, u τ, and ũ τ satisfy te conditions of Lemma 42, Remark 43, and Lemma 61 Ten te error between te solution u of problems (21), (22) and te solution u τ of problems (31), (32) is bounded from above by u u τ W r (,T ;W 1,ρ (Ω),W 1,π (Ω)) N c u π u 1,π + τ n [(η) n p + rτ n p 1,π ] N N + τ n (Θ n ) p + τ n u n u n 1 r 1,ρ } 2/r } (71) and from below by N τ n [(η) n p + rτ n p 1,π ] c u u τ W r (,T ;W 1,ρ (Ω),W 1,π (Ω)) N N + τ n (Θ n ) p + τ n u n u n 1 r 1,ρ } 2/r } (72) Te quantities η n, Θn and rn τ are defined in equations (56), (57) and (61) respectively Proof Te upper bound (71) immediately follows from te decomposition (46) of te residual and Lemmas 42, 53 and 61 In view of Lemma 42, te lower bound (72) is establised once we ave proved tat N τ n [(η) n p + rτ n p 1,π ] c u τ + G(u τ ) Lp (,T ;W 1,π (Ω)) N N } 2/r } + τ n (Θ n ) p + τ n u n u n 1 r 1,ρ (73) 17

18 We start wit te r n τ -term on te left-and side of estimate (73) From te decomposition (46) and Lemmas 53 and 61 we obtain N τ n r n τ p 1,π 2p + 1 R τ (u τ ) Lp (,T ;W 1,π (Ω)) + γ N } 2/r } τ n u n u n 1 r 1,ρ 2 2p + 1 u τ + G(u τ ) L p (,T ;W 1,π (Ω)) N + c τ n [(η) n p + (Θ n ) p ] + γ 2 N τ n u n u n 1 r 1,ρ } 1/ρ } 2/r } (74) To bound te η n -term on te left-and side of estimate (73) we set w = N (α + 1)( t t n 1 ) α (η) n p π w n χ (tn 1,t n ] (t), τ n were te functions w n are as in Lemma 53 and χ (tn 1,t n ] denotes te caracteristic function of te interval (t n 1, t n ] Te parameter α is arbitrary at present and will be fixed later Since ( t t n 1 τ n ) α 1 on [t n 1, t n ] and since (p 1)p = p, we obtain from te second line of estimate (59) tat w L p (,T ;W 1,π c (α + 1) = c (α + 1) N (Ω)) N τ n (η) n (π 1)p (η) n (p π)p } p 1 p τ n (η) n p (75) Since tn t n 1 (α + 1)( t t n 1 τ n ) α dt = τ n, 18

19 te first line of estimate (59) implies tat N τ n (η) n p T T R (u τ ), w + Equations (46) and (64) yield T Estimate (75) gives T u τ N τ n Θ n c (η) n π 1 (η) n p π N N R (u τ ), w + c τ n (Θ n ) p R (u τ ), w = u τ τ T u τ + G(u τ ), w + G(u τ ), w Estimates (67) and (75) imply γ 2 T R τ (1) (u τ ), w + G(u τ ) Lp (,T ;W 1,π (Ω))c (α + 1) R (2) τ (u τ ), w N τ n u n u n 1 r 1,ρ} 2/r c (α + 1) T N } p 1 p τ n (η) n p R τ (2) (u τ ), w N Equation (68) and te second line of estimate (59) finally yield = = T R (1) τ (u τ ), w N rτ n, w n (η) n p π (α + 1) tn N rτ n, w n (η) n p π (θ α + 1 α + 2 )τ n c θ α + 1 N α + 2 τ n rτ n 1,π (η) n p 1 c θ α + 1 N α + 2 τ n rτ n p 1,π (θ t t n 1 t n 1 τ n N } p 1 p τ n (η) n p } p 1 p τ n (η) n p )( t t n 1 ) α dt τ n } p 1 p τ n (η) n p (76) (77) (78) (79) (71) 19

20 Equation (77) and estimates (76), (78), (79), and (71) give N τ n (η) n p c (α + 1) u τ + G(u τ ) L p (,T ;W 1,π (Ω)) + c (α + 1) γ N } 2/r τ n u n u n 1 r 1,ρ 2 N + c τ n (Θ n ) p + c θ α + 1 N α + 2 τ n r n τ p 1,π Inserting estimate (74) in tis inequality we finally arrive at N τ n (η) n p c [(α + 1) + θ α + 1 α + 2 2p + 1}1/p ] u τ + G(u τ ) Lp (,T ;W 1,π (Ω)) + c [1 + θ α + 1 N α + 2 c 2p + 1 ] τ n (Θ n ) p + γ 2 c [(α + 1) + θ α + 1 α + 2 2p + N 1}1/p ] + c c θ α + 1 N 2p + 1}1/p τ n (η n α + 2 ) p τ n u n u n 1 r 1,ρ } 2/r (711) Now we coose te parameter α suc tat te η n -term on te rigt-and side of estimate (711) is balanced by te term on te left-and side For te mid-point rule, ie θ = 1 2, tis is obvious: We simply coose α = and te ηn -term of te rigt-and side of (711) vanises In te case 1 2 < θ 1 we coose α = Tis coice ensures tat 2K(2θ 1) 2K(1 θ) + 1 wit K = c 2p + 1 c c θ α + 1 α + 2 2p + 1}1/p and α 2K

21 Hence, estimate (711) takes te form N τ n (η) n p c u τ + G(u τ ) Lp (,T ;W 1,π (Ω)) (712) N + τ n (Θ n ) p + τ n u n u n 1 r 1,ρ } 2/r wit a constant c tat only depends on p, c and c Estimates (74) and (712) prove inequality (73) and tus complete te proof of te lower bound (72) 8 W 1,q -stability results for te Laplacian Te results of te next section are based on te W 1,q -stability of te Laplacian bot in its analytical and discrete form We start wit te analytical case Te following result is well-known for domains wit smoot C 1 -boundary (cf [8]) For polygonal domains, owever, we are not aware of a proof Recall tat for q [1, ] te dual Lebesgue exponent is denoted by q [1, ] and is defined by 1 q + 1 q = 1 81 Lemma For every convex, bounded, polygonal domain Ω IR 2 and every q [1, ] tere is a constant α q > suc tat inf v W 1,q (Ω) sup w W 1,q (Ω) Ω v w v,q w,q α q (81) Te constant α q only depends on q and on te maximum interior angle at te vertices of Ω Proof Inequality (81) is proved in [8] for domains Ω IR n, n 2, wit smoot C 1 -boundary Te proof is based on te following tree auxiliary results: 1 Inequality (81) olds for IR n 2 Inequality (81) olds for te alf-space H + = x IR n : x 1 > } 3 Inequality (81) olds for domains H ω = x IR n : x 1 > ω(x 2,, x n )} wit functions ω C 1 (IR n 1 ) satisfying ω() = and ω L (IR n 1 ) 1 Te tird result is te only point were te smootness of te boundary comes into play Te smootness condition on ω can be relaxed to te condition tat ω sould be Lipscitz continuous and tat its Lipscitz constant is sufficiently small Tis, owever, does not elp us since it would require tat te interior angles at te 21

22 vertices of Ω sould be sufficiently close to π Instead we must prove tat inequality (81) olds for domains H c = x IR 2 : x 2 cx 1 } wit c > To verify tis, coose a parameter c > and keep it fixed Ten introduce polar coordinates to transform H c to te strip (r, ϕ) : r >, ϕ α} were α = arctan c Next apply te scaling r 2α π r, ϕ 2α π ϕ to transform to te strip (s, ψ) : s >, ϕ π 2 } Ten transform back to cartesian coordinates Te combination of tese transformations transforms H c to te alf space H + Now, we already know tat inequality (81) olds for H + Hence it also olds in polar-coordinates Since te left-and side of (81) is invariant under scalings, inequality (81) olds in polar coordinates on te strip (r, ϕ) : r >, ϕ α} and tus on H c Once we know tat we may replace H ω by H c, te rest of te proof of te lemma proceeds as in [8] 82 Remark Wen Ω is not convex but as a re-entrant corner wit angle ω > π, 2ω Lemma 81 can at best old for Lebesgue exponents q [1, ω π ) Tis is due to te fact tat te singular solution r π ω sin( π ω ϕ) of te Laplacian is in W 1,q (Ω) only for tis realm of Lebesgue exponents In te proof of Lemma 81 te convexity is reflected by te fact tat te transformation from polar to cartesian coordinates is globally invertible in te vicinity of convex corners For non-convex corners it is only locally invertible Now we come to te discrete case 83 Lemma Consider a convex, bounded, polygonal domain Ω IR 2 and an arbitrary affine equivalent, admissible and sape regular partition T of Ω Denote by S 1 (T ) = v C(Ω) : v K R 1 (K) K T, v = on T } (82) te space of continuous piecewise linear finite element functions corresponding to T Ten for every q [1, ] tere is a constant β q > suc tat Ω inf sup v T w T β q (83) v T,q w T,q v T S 1 (T ) w T S 1 (T ) Te constant β q only depends on q, on te maximum interior angle at te vertices of Ω, and on te sape parameter sup K T K /ρ K of T Proof We denote by R T : W 1,1 (Ω) S 1 (T ) te Ritz projection wic is defined by Ω (R T v) w T = Ω v w T v W 1,1 (Ω), w T S 1 (T ) Consider first te case q [1, 2] Ten we ave q 2 From [7] and [4, Cap 7] we know tat R T is stable in te W 1,q -norm, ie, tere is a constant c q > suc tat (R T w),q c q w,q w W 1,q (Ω) 22

23 Te constant c q only depends on q, on te maximum interior angle at a vertex of Ω, and on te sape parameter of T Consider an arbitrary function v T S 1 (T ) and a number δ (, 1) From Lemma 81 we know tat tere is a function w δ W 1,q (Ω) wit w δ,q = 1 and v T w δ = δα q v T,q Ω Togeter wit te stability of te Ritz projection tis implies Ω sup v T w T Ω v T (R T w δ ) w T S 1 (T ) v T,q w T,q v T,q (R T w δ ),q 1 Ω v T (R T w δ ) c q v T,q w δ,q = 1 Ω v T w δ c q v T,q w δ,q δα q c q Since δ and v T were arbitrary tis proves inequality (83) wit β q = α q c q One easily cecks tat (83) implies te stability of R T in te W 1,q -norm wit c q = 1 β q = c q α q Now consider te case q > 2 Tis implies 1 < q < 2 Since we already ave establised te stability of R T in te W 1,q -norm wit c q = 1 β q = c q α q in te case q 2 and obtain inequality (83) wit β q = α q c q = α qα q c q, we can proceed as 9 Te final a posteriori error estimate In tis section we make computable te error estimator of Lemma 71 by replacing te negative Sobolev norms of te r n τ -terms by computable quantities Tis will be done wit te elp of suitable auxiliary discrete Poisson equations As in Section 5 we coose an integer l and denote for every n between 1 and N by a p;,n (x, u nθ, u nθ ), a u;,n (x, u nθ, u nθ ), b p;,n (x, u nθ, u θ ), b u;,n (x, u nθ, u nθ ) te L 2 -projections of a p (x, u nθ, u nθ ), a u (x, u nθ, u nθ ), b p (x, u nθ, u nθ ), b u (x, u nθ, u nθ ) on discontinuous tensors, vector-fields and functions respectively wic are piecewise polynomials of degree l on te elements of T,n For abbreviation we set for every 23

24 element K T,n, every edge E Ẽ,n and every n between 1 and N R K = div[a p;,n (x, u nθ, u nθ ) (u n u n 1 )] div[a u;,n (x, u nθ, u nθ )(u n u n 1 )] + b p;,n (x, u nθ, u nθ ) (u n u n 1 ) + b u;,n (x, u nθ, u nθ )(u n u n 1 ), R E =a p;,n (x, u nθ, u nθ ) (u n u n 1 ) + a u;,n (x, u nθ, u nθ )(u n u n 1 ), D K = div[(a p (x, u nθ, u nθ ) a p;,n (x, u nθ, u nθ )) (u n u n 1 )] div[(a u (x, u nθ, u nθ ) a u;,n (x, u nθ, u nθ ))(u n u n 1 )] + (b p (x, u nθ, u nθ ) b p;,n (x, u nθ, u nθ )) (u n u n 1 ) + (b u (x, u nθ, u nθ ) b u;,n (x, u nθ, u nθ ))(u n u n 1 ), D E =(a p (x, u nθ, u nθ ) a p;,n (x, u nθ, u nθ )) (u n u n 1 ) + (a u (x, u nθ, u nθ ) a u;,n (x, u nθ, u nθ ))(u n u n 1 ) Of course, te rigt-and sides of te above equations must always be interpreted as te restriction of corresponding functions to te relevant element or edge Recall te definitions (61) of te residuals r n τ and (82) of te spaces S 1 (T ) 91 Lemma For every integer n between 1 and N denote by ũ n S1 ( T,n ) te unique solution of te discrete Poisson equation Define te error indicator η n by η n = Ω and te data error Θ n by ũ n v = r n τ, v v S 1 ( T,n ) (91) K T,n π K R K + ũ n π,π;k + E Ẽ,n E [n E ( ũ n R E )] E π π;e Θ n = π K D K π,π;k + E D E π π;e K T,n E Ẽ,n 24 } 1/π (92) 1/π (93)

25 Ten tere are two constants c and c, wic only depend on te polynomial degree l and on te ratios K /ρ K, suc tat r n τ 1,π η n + ũ n,π c η n + ũ n,π + Θ n }, c r n τ 1,π + Θ n } (94) Proof We coose an integer n between 1 and N and keep it fixed Lemma 81 implies tat te Poisson equation Ũ n v = rτ n, v v W 1,π (Ω) Ω admits a unique solution Ũ n W 1,π (Ω) and tat Ũ n,π 1 α π r n τ 1,π Te definition of te negative Sobolev norms on te oter and yields Lemma 83 similarly gives Te triangle inequality terefore implies r n τ 1,π Ũ n,π ũ n,π 1 β π r n τ 1,π 1 3 minα π, β π } ũ n,π + (Ũ n ũ n ),π } r n τ 1,π ũ n,π + (Ũ n ũ n ),π (95) Using standard arguments (cf eg [9]) we infer from Lemmas 51 and 52 tat (Ũ n ũ n ),π c η n + Θ n }, η n C (Ũ n ũ n ),π + Θ n } (96) wit constants c and C wic only depend on te polynomial degree l and te ratios K /ρ K Combining estimates (95) and (96) we arrive at te desired estimate (94) of r n τ 1,π A standard duality argument for te L 2 -projection onto finite element spaces yields 1/π u π u 1,π c π K u π u π,π;k K T, Combining tis estimate and Lemmas 71 and 91 proves our final result: 25

26 92 Teorem If te conditions of Lemma 71 are satisfied te error between te solution u of problems (21), (22) and te solution u τ of problems (31), (32) is bounded from above by u u τ W r (,T ;W 1,ρ (Ω),W 1,π (Ω)) N c τ n [(η) n p + ( η ) n p + ũ n p,π ] N + τ n [(Θ n ) p + ( Θ n ) p ] + π K u π u π,π;k K T n, N } 2/r } + τ n u n u n 1 r 1,ρ 1/π (97) and from below by N τ n [(η) n p + ( η ) n p + ũ n p,π ] c u u τ W r (,T ;W 1,ρ (Ω)) N + τ n [(Θ n ) p + ( Θ n ) p ] + N n+1 τ n u n u n 1 r 1,ρ } 2/r } (98) Te quantities η n, Θn, ηn, Θ n, and ũn (93), and (91) respectively are defined in equations (56), (57), (92), 93 Remark Te left-and side of estimate (98) is our error indicator Its first term controls te error of te space-discretization and can be used for adapting te spatial mes Te second and tird terms on te left-and side of (98) control te error of te time-discretization and can be used to adapt te temporal mes Te last terms on te rigt-and sides of estimates (97) and (98) are not present for linear differential equations Tey control te linearization error tat is implicit in te discretization Since tey are computable, tey can be used to control tis linearization error Tese contributions are of ig-order in te sense tat up to a 26

27 different L p -norm, tey are similar to te square of te error indicator Te second and tird term on te rigt-and side of estimate (97) and te second term on te rigt-and side of estimate (98) are data errors In contrast to linear problems tey not only involve given data but te discrete solution as well 94 Remark Estimate (98) is based on te lower bound of Lemma 42 Tis in turn follows from te lower bound in te abstract error estimate of Lemma 41 Te latter only involves te Frécet derivative DF (u) Applied to differential equations tis corresponds to a linearized differential operator Since suc operators ave a local effect, te lower bounds can be localized Terefore, as for linear differential equations (cf [12]), estimate (98) as an analogue tat is local wit respect to time 1 References [1] R A Adams: Sobolev Spaces Academic Press, New York, 1975 [2] H Amann: Linear and Quasilinear Parabolic Problems Volume I: Abstract Linear Teory Birkäuser, Basel, 1995 [3] G Bangert, R Rannacer: Adaptive Finite Element Metods for Differential Equations Birkäuser, Basel, 23 [4] S C Brenner, L R Scott: Te Matematical Teory of Finite Element Metods Springer, Berlin - Heidelberg - New York, 1994 [5] P Clément: Approximation by finite element functions using local regularization RAIRO Anal Numér 9, (1975) [6] R Dautray and J-L Lions: Matematical Analysis and Numerical Metods for Science and Tecnology Springer, Berlin - Heidelberg - New York, 1992 [7] R Rannacer, L R Scott: Some optimal error estimates for piecewise linear finite element approximations Mat Comput 38, (1982) [8] C G Simader, H Sor: Te Diriclet Problem for te Laplacian in Bounded and Unbounded Domains Longman, London, 1996 [9] R Verfürt: A Review of A Posteriori Error Estimation and Adaptive Mes-Refinement Tecniques Teubner-Wiley, Stuttgart, 1996 [1] : A posteriori error estimates for nonlinear problems: L r (, T ; W 1,ρ (Ω)) error estimates for finite element discretizations of parabolic equations Numer Met for PDE 14, (1998) [11] : Error estimates for some quasi-interpolation operators Modél Mat et Anal Numér 33, (1999) [12] : A posteriori error estimates for finite element discretizations of te eat equation Calcolo 4, (23) 27