EXPLICIT ERROR ESTIMATES FOR COURANT, CROUZEIX-RAVIART AND RAVIART-THOMAS FINITE ELEMENT METHODS

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1 EXPLICI ERROR ESIMAES FOR COURAN, CROUZEIX-RAVIAR AND RAVIAR-HOMAS FINIE ELEMEN MEHODS CARSEN CARSENSEN, JOSCHA GEDICKE, AND DONSUB RIM Abstract. he elementary analysis of this paper presents explicit expressions of the constants in the a priori error estimates for the lowest-order Courant, Crouzeix-Raviart nonconforming and Raviart-homas mixed finite element methods in the Poisson model problem. he three constants and their dependences on some maximal angle in the triangulation are indeed all comparable and allow accurate a priori error control. 1. Introduction Quantitative a priori error control for the three most popular lowest-order conforming, nonconforming, and mixed 2D finite element methods (FEMs) named after Courant, Crouzeix-Raviart, and Raviart-homas, depicted symbolically in Figure 1.1, is one of the most fundamental questions in the numerical analysis of partial differential equations (PDEs). For the Courant FEM and the Raviart-homas mixed FEM (MFEM), there exist elementwise interpolation operators I and I F such that the error analysis consists in an estimate of the Lebesgue norms in the sense of (v Iv) L 2 ( ) C( )h D 2 v L 2 ( ) for some smooth function v with Hessian D 2 v and the triangle with diameter h. he point is that the constant C( ) depends on the shape of the triangle but not on its size h. he textbook analysis is based on the Bramble-Hilbert lemma and so on some compact embeddings on a reference geometry [Bra01, Cia78]. he transformation formula then leads to some estimate of C( ) which is qualitative and can be quantified with the help of computer-justified values of some eigenvalue problem on the reference triangle, cf. e.g. [KL07] for a historic overview and the references quoted therein, in particular [BA76] for Courant his research was supported by the World Class University (WCU) program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and echnology R and the DFG Research Center MAHEON Mathematics for key technologies in Berlin. he second author was supported in parts by the DFG graduate school BMS Berlin Mathematical School.

2 2 C. CARSENSEN, J. GEDICKE, AND D. RIM Figure 1.1. Courant, Crouzeix-Raviart, and Raviart-homas FE. and [AD99] for Raviart-homas FEM. his paper aims at direct elementary proofs of quantitative error estimates based on the Poincaré inequality with some known constant plus elementary integration by parts. he situation is somewhat different for the nonconforming FEMs because the local interpolation error through the natural interpolation operator I NC is very sharp, even optimal by some averaging property; but the global error is also driven by the interaction with the inconsistency. he standard textbook analysis employs some Strang-Fix type argument [Bra01, BS08] which leads to two contributions and gives the reader the impression that the error analysis is even more sensitive and perhaps even the scheme is more sensitive than the other two. In Braess [Bra01] page 111 one can even find the hint that the Crouzeix-Raviart nonconforming FEM (NCFEM) is more sensitive with respect to large second order derivatives than the other two methods. his paper aims at a clarification by the comparison of the best known constants C( ) for the three FEMs at hand. In fact, the constant 1/4 + 2/j1,1 2 (1.1) C(α) := 1 cos α, for a maximal angle 0 < α < π of a triangle and the first positive root j 1,1 of the Bessel function J 1, and its maximum C( ) := max C(max ) in a triangulation of a 2D polygonal domain Ω play a dominant role. he main results of this paper are the explicit error estimates (1.2) u u C C( ) h D 2 u, L 2 (Ω) (1.3) p p R C( ) h L2(Ω) Dp, L2(Ω) (1.4) u u CR NC 1 osc(f, ) + j 1,1 1 j 2 1,1 + C( ) 2 h D 2 u L 2 (Ω)

3 EXPLICI ERROR ESIMAES FOR FEM 3 for the Courant, Raviart-homas and Crouzeix-Raviart finite element approximations u C, p R and u CR in a simple Poisson model problem and the oscillations osc(f, ) defined in Section 6. In particular, the constants (which are upper bounds) have the same behaviour as the angles deteriorate with α π. he above estimate for the NCFEM displays the perturbation result for an arbitrary L 2 function f as a right-hand side in the Poisson model problem and thereby corrects and sharpens a corresponding error analysis in [MS09]. he technique here bypasses the Strang-Fix argument by the direct connection of the Raviart-homas MFEM with the Crouzeix-Raviart NCFEM usually attributed to Marini [Mar85, AB85]. he paper is organised as follows. Section 2 presents some preliminaries and Section 3 shows the elementary interpolation estimate for the nodal interpolation operator I. he model problem and the error estimate (1.2) for the Courant finite element method is presented Section 4. Sections 5 and 6 present the error estimates for the Raviart-homas MFEM (1.3) and the Crouzeix-Raviart NCFEM (1.4). he contents of this paper reflects the way, finite element methods are taught by the first author over the years at the universities in Kiel, Vienna, Berlin, Budapest, and Seoul as well as in his summer schools in Cape own, Beijing, Mumbai and on Goa. hey seem to be optimal in the class of arguments and offer some quantitative insight with surprisingly little effort. hroughout this paper, standard notation on Lebesgue and Sobolev spaces is employed. he Lebesgue integral reads, the integral mean ffl, norms := L 2 (Ω), NC := NC L 2 (Ω) with piecewise gradient ( NC ) := ( ) for all, and denotes the measure as the area of the triangle and the length E of an edge E. 2. Elementary Preliminaries his section is devoted to some preliminaries for the interpolation error estimates. One is a Poincaré-Friedrichs type estimate, which follows from the well known trace identity and another is some transformation stability in the plane. Figure 2.1 displays the geometry of a triangle in the subsequent two lemmas. Lemma 2.1 (race Identity). Let f W 1,1 ( ) on the triangle = conv({p } E) with vertex P and opposite edge E. hen it holds E f ds f dx = 1 2 (x P ) f(x) dx.

4 4 C. CARSENSEN, J. GEDICKE, AND D. RIM H ν E E ν x P Figure 2.1. Geometry of the triangle from Lemma 2.1 and Lemma 2.3. Proof. Set g(x) := (x P )f(x) for all x and observe (x P ) ν E = dist(p, H) for x E H for the line H from Figure 2.1 that enlarges E. For x on one of the two other edges, x P is parallel to that edge. Hence, the unit normal ν along satisfies (x P ) ν(x) for x \ E. herefore, the Gauss divergence theorem leads to div g(x) dx = g(x) ν(x) ds x = f(x) (x P ) ν E ds x = dist(p, H) his and the product rule prove the assertion. E div g(x) = 2f(x) + f(x) (x P ) E f ds. he classical Poincaré constant of Payne-Weinberger [PW60] has recently been improved from 1/π (for all convex domains) to the optimal value 1/j 1,1 (for triangles), where j 1, denotes the first positive root of the Bessel function J 1. heorem 2.2 (Poincaré Inequality on riangles [LS10]). For all f H 1 ( ) on a triangle it holds (2.1) f ffl f(x)dx h L 2 ( ) /j 1,1 f H 1 ( ).

5 EXPLICI ERROR ESIMAES FOR FEM 5 Lemma 2.3 (Poincaré-Friedrichs Inequality). Let f H 1 ( ) on the triangle = conv({p } E) with vertex P opposite of the edge E satisfy f ds = 0. E hen it holds f L 2 ( ) max x E P x 2 /8 + h 2 /j2 1,1 f H 1 ( ). Proof. he theorem of Pythagoras for a := f ffl f(x)dx and b := ffl f(x)dx reads f 2 L 2 ( ) = a + b 2 L 2 ( ) = a 2 L 2 ( ) + b 2 L 2 ( ). he Poincaré inequality (2.1) gives a L 2 ( ) = f ffl f(x)dx L 2 ( ) h /j 1,1 f H 1 ( ). he trace identity from Lemma 2.1 with ffl f ds = 0 leads to E b = f(x) dx = 1 (x P ) f(x) dx P L 2 ( ) f H 1 ( ). With polar coordinates (r, ϕ) and the notation for x P =: r and α < ϕ < β with some distance 0 < δ(ϕ) max x E P x of P to E, one deduces x P 2 L 2 ( ) = β α δ(ϕ) 0 r 2 r dr dϕ = max x E P x 2 /2 β α β δ(ϕ) 0 0 δ(ϕ) 4 /4 dϕ r dr dϕ = max x E P x 2 /2. his results in the bound b = f(x) dx 2 3/2 1/2 max P x f H x E 1 ( ). he preceding two estimates control the two terms a and b of the above Pythagoras identity and so prove f 2 L 2 ( ) h2 /j1,1 2 f 2 H 1 ( ) + max P x E x 2 /8 f 2 H 1 ( ) ( ) = max P x E x 2 /8 + h 2 /j1,1 2 f 2 H 1 ( ). he following inequality compares the Euclidean length a of a vector a in the plane with a second metric (a ν) 2 + (a µ) 2 given by the two projections a ν and a µ. Lemma 2.4 (ransformation Stability). For linearly independent unit vectors ν and µ in R 2, it holds (a ν) 2 + (a µ) 2 min = 1 ν µ a R 2 \{0} a 2

6 6 C. CARSENSEN, J. GEDICKE, AND D. RIM Proof. Let a = αν + βµ for real α and β with α 2 + β 2 = 1. Set γ := ν µ and ν = 1 = µ. hen 1 2αβ 1 and so his is equivalent to 0 (1 + γ )( γ + 2αβγ). (1 + γ ) 2αβγ γ 2 + γ. Add 1 γ + 4αβγ on both sides to prove LHS := (1 γ )(1 + 2αβγ) = 1 + 2αβγ γ 2αβγ γ Direct calculations show 1 + γ 2 + 4αβγ =: RHS. a 2 = α 2 + β 2 + 2αβγ = 1 + 2αβγ = LHS/(1 γ ) as well as a ν = α + βγ and a µ = β + αγ. herefore Altogether this proves, (a ν) 2 + (a µ) 2 = (α + βγ) 2 + (β + αγ) 2 = 1 + γ 2 + 4αβγ = RHS. (1 γ ) a 2 (a ν) 2 + (a µ) 2 for all a R 2. his shows that the left-hand side in the assertion is in fact larger than or equal to 1 γ. Equality and attainment of the minimum follows with the choice (α, β) = 1 2 (±1, 1) for ± γ 0 plus direct calculations. his concludes the proof. 3. Nodal Interpolation Error Estimate his section presents the nodal interpolation error estimates in an abstract form on a triangle with focus on explicit constants and then compares with the estimate from [KL07]. Recall the expression C(α) from (1.1) for any angle 0 < α < π of a triangle which is preferably chosen as C(max ). heorem 3.1 (Interpolation Error Estimate). Let v H 2 ( ) with v(a) = v(b) = v(c) = 0 on the triangle = conv{a, B, C}, with vertices A, B, C, diameter h, and some interior angle 0 < α < π. hen it holds v L 2 ( ) C(α) h D 2 v. L 2 ( )

7 EXPLICI ERROR ESIMAES FOR FEM 7 C τ 2 B α τ 1 A Figure 3.1. Geometry in heorem 3.1. Proof. Figure 3.1 displays two unit vectors τ 1 = ν and τ 2 = µ along the two sides of the angle α with γ := τ 1 τ 2 = cos α. Lemma 2.4 for a := v(x) and f j := τ j v(x) plus integration over show ( (1 γ ) v(x) 2 dx f1 (x) 2 + f 2 (x) 2) dx. Lemma 2.3 proves ( f1 (x) 2 + f 2 (x) 2) dx max{ A B 2, A C 2 }/8 + h 2 /j 2 1,1 ( ) f 1 2 H 1 ( ) + f 2 2 H 1 ( ). Since τ j = 1 it holds for all x and j, k = 1, 2 that f j (x) = v(x) τ j 2 2 v(x) x k x k x k x l his and l=1 2. f j 2 H 1 ( ) = f j/ x 1 2 L 2 ( ) + f j/ x 2 2 L 2 ( ) for j = 1, 2 (which eventually results in the factor 2) lead to v 2 H 1 ( ) 1/4 + ( 2/j2 1,1 h v 1 cos α v x 1 x v 2) dx x 2 1 x 2 2 = 1/4 + 2/j2 1,1 1 cos α h 2 v 2 H 2 ( ) = C(α)2 h 2 v 2 H 2 ( ). he following example illustrates how the estimate has to deteriorate as α π and why it stays bounded under the maximal angle condition.

8 8 C. CARSENSEN, J. GEDICKE, AND D. RIM (0, δ) (0, δ) 1 ( 1, 0) (1, 0) 2 (0, 0) (1, 0) Figure 3.2. he triangles 1 and 2 from Example 3.2. Example 3.2 (Maximal Angle Condition). Given any 0 < δ 1, consider the triangles 1 and 2 defined in Figure 3.2 with vertices N ( 1 ) and N ( 2 ). he point is that 2 has some largest angle α = π/2 while that of 1 is α = 2 arctan(1/δ) and this tends to π as δ tends to zero. he smooth function v(x 1, x 2 ) = 1 x 2 1 has the nodal interpolation Iv(x 1, x 2 ) = x 2 /δ on 1 and one calculates (v Iv)/ x 2 2 L 2 ( 1 ) = 1/δ (v Iv) 2 L 2 ( 1 ) for D 2 v 2 L 2 ( 1 ) = 4δ. heorem 3.1 applies to the interpolation error v Iv as it vanishes at the vertices of 1. his shows Q(v) := (v Iv) L 2 ( 1 ) h 1 D 2 v L 2 ( 1 ) Elementary trigonometric considerations show C(α). (1 cos α ) 1 = (1 + δ 2 )/(2δ 2 ) δ 2. Hence, the lower and upper bounds show the same asymptotic behaviour (4δ) 1 Q(v) δ 1 1/4 + 2/j 2 1,1 as δ 0 In other words, the degeneracy of C(α) is sharp in the sense that sup v H 2 ( 1 )\{0} v=0 at N ( 1 ) Q(v) (1 cos α ) 1/2 as α π. o illustrate the difference to the triangle 2 with right angle α = π/2, note that C(π/2) = 1/4 + 2/j1, is bounded independently of δ 0. he search of an optimal bound the error estimate of heorem 3.1 can also be posed as an eigenvalue problem with the Rayleigh quotient RQ(v) := v 2 L 2 ( ) / D 2 v 2 L 2 ( ) for v H 2 ( ) with v = 0 on N ( )

9 EXPLICI ERROR ESIMAES FOR FEM 9 heorem 3.1 leads to an upper bound of the first eigenvalue of this eigenvalue problem with an elementary proof. he value C 3 = is known for a right isosceles triangle from [Arb82, Leh86, Sig88]. Remark 3.3 (Comparison with [KL07]). he reference [KL07] discusses a valid upper bound for the constant C 3 ( ) 2 := sup v H 2 ( )\{0} v=0 at N ( ) RQ(v) C(α) 2 h 2, for a triangle with maximal angle α and diameter h = diam( ). Based on some transformation arguments, this constant has been computed and empirically studied in [KL07] and formerly in [Arb82, Leh86, Sig88] with the numerical value C 3 ( ref ) = on the reference triangle ref = conv{(0, 0), (0, 1), (1, 0)}. Justified by computer-simulations, the bound of [KL07] leads to (3.1) C 3 ( ) 1 + cos α 2 1 cos α C 3 ( ref ) 1 cos α h. (he reader is warned that the notation in [KL07] is different and does not involve the maximum length h but the second largest one and there is another parameter which is maximized here in (3.1) for simplicity.) Figure 3.3 compares the upper bound in (3.1) for C 3 ( ) and the bound C(α) as a function of the angle α in the range π/3 α < π. Notice that an equilateral triangle with α = π/3 shows C(α) = < C 3 ( ) = and the bound of heorem 3.1 is even sharper than that of [KL07]. his is not a contradiction because the transformation in [KL07] leads to some upper bound. he overall conclusion from Figure 3.3 is that the two bounds are comparable; one is with an elementary proof, while the other is justified by numerical calculations. 4. Courant FEM his section is devoted to the simplest model problem for second-order elliptic PDEs and its most elementary first-order conforming discretisation Poisson Model Problem. Given a right-hand side f L 2 (Ω) on a bounded Lipschitz domain Ω R 2 with polygonal boundary Ω, the strong form of the Poisson model problem reads: seek u C( Ω) Hloc 2 (Ω) such that (4.1) u = f in Ω and u = 0 along Ω.

10 10 C. CARSENSEN, J. GEDICKE, AND D. RIM upper bound for C 3 () C( ) Figure 3.3. Comparison of the constant C(α) and the upper bound (3.1) for C 3 ( ). he formally equivalent weak formulation utilizes the scalar product and the linear and bounded functional a(u, v) := u v dx and F (v) := fv dx for u, v V := H0(Ω) 1 Ω in the Hilbert space H 1 0(Ω) of Lebesgue measurable functions in L 2 (Ω) with a weak gradient in L 2 (Ω; R 2 ). he weak form seeks the Riesz representative u of F within the Hilbert space V, namely u V with (4.2) a(u, v) = F (v) for all v V. Elliptic regularity leads to u H 2 loc (Ω) H1+s (Ω) for some 1/2 < s 1 with s = 1 for convex domains [Eva10, G01] Regular riangulation. A regular triangulation of Ω (in the sense of Ciarlet) into triangles is a finite set of closed triangles of positive area such that := = Ω and any two distinct triangles 1 and 2 in with 1 2 share exactly one vertex z or have one edge E in common. he set of all edges of a triangle is denoted by E( ), the set of vertices of is denoted by N ( ). he set of all edges resp. nodes is written as E := E( ) and N := N ( ). Let mid(e) := {mid(e) E E} be the set of midpoints of the edges. Ω

11 EXPLICI ERROR ESIMAES FOR FEM 11 he piecewise constant weight h P 0 ( ) is the local mesh-size, h := h := diam( ) for all Courant FEM. For a regular triangulation of Ω and k N 0 define the finite element spaces P k ( ) := {polynomial on with degree k}, P k ( ) := {v L 2 (Ω), v P k ( )}, V C ( ) := C 0 (Ω) P k ( ). he nodal basis function ϕ z C(Ω) P 1 ( ) is defined by ϕ z (z) = 1 and ϕ z (y) = 0 for z N and all other nodes y N \ {z}. he nodal interpolant is the operator I : C(Ω) V C ( ), v z N v(x)ϕ z (x). he Galerkin discretisation replaces H 1 0(Ω) by the finite element space V C ( ): seek u C V C with (4.3) a(u C, v C ) = F (v C ) for all v C V C. he following immediate consequence of heorem 3.1 and the well-known optimality of u C implies (1.2). Corollary 4.1. he Courant FEM solution u C on Ω of the Poisson model problem (4.1) satisfies u u C u Iu C( ) h D 2 u L 2 (Ω). Proof. he first inequality follows from Galerkin orthogonality and the second from heorem 3.1 because u Iu vanishes at all nodes Numerical Example. Consider the Poisson model problem (4.1) with f(x, y) = 4 2x 2 2y 2 for (x, y) Ω := ( 1, 1) 2 and exact solution u(x, y) = (1 x 2 )(1 y 2 ). he sequence of uniform criss triangulations ( l ) l of the unit square Ω is generated by uniform refinements of Ω into squares divided along the diagonal parallel to the main diagonal. able 4.1 shows the errors computed with Matlab [ACF99] for different levels l with mesh-sizes h l = 2/2 l and efficiency indices EI := (C( l ) hl D 2 u )/ u u L 2 ( l ) C.

12 12 C. CARSENSEN, J. GEDICKE, AND D. RIM l u u C u Iu EI able 4.1. Numerical results for Courant FEM. 5. Raviart-homas MFEM his section is devoted to the error analysis of the Raviart-homas mixed finite element method. he first subsection presents the key argument Fortin Interpolation Error Estimate. his subsection is devoted to the error analysis in simplified notation. he Fortin interpolation will be defined in Subsection 5.2 below. heorem 5.1 (Fortin Interpolation Error Estimate). Let q H 1 ( ; R 2 ) on the triangle = conv{p 1, P 2, P 3 } with maximal angle α with q ν E ds = 0 for all E E( ). hen it holds E q L 2 ( ) C(α) h Dq L 2 ( ). Proof. Let E 1, E 2, E 3 be the edges of and ν 1, ν 2, ν 3 corresponding exterior unit normal vectors. hen f j := q ν j H 1 ( ) satisfies f j ds = 0 E j for any j = 1, 2, 3. Lemma 2.3 implies f j L 2 ( ) h 1/8 + 1/j1,1 2 f j H 1 ( ). Suppose that the maximum angle α of is at P 3 with neighbouring edges E 1 and E 2. Lemma 2.4 implies for ν = ν 1 and µ = ν 2 with ν 1 ν 2 = cos α that (1 cos α ) q 2 L 2 ( ) f 1 2 L 2 ( ) + f 2 2 L 2 ( ) Since ν j is a unit vector, (5.1) Hence h 2 (1/8 + 1/j 2 1,1)( f 1 2 H 1 ( ) + f 2 2 H 1 ( ) ). f j H 1 ( ) = Dq ν j L 2 ( ) Dq L 2 ( ). f 1 2 H 1 ( ) + f 2 2 H 1 ( ) 2 q 2 H 1 ( ).

13 EXPLICI ERROR ESIMAES FOR FEM 13 proves the as- he combination with the aforementioned estimate of q L 2 ( ) sertion Raviart-homas Finite Element Space. Given a regular triangulation from Subsection 4.2, define the Raviart-homas finite element space R 0 ( ) := {q R P 1 ( ; R 2 ) H(div, Ω) : a, b, c R x, q R (x) = (a, b ) + c (x 1, x 2 )}. It is well known that some piecewise polynomial function q R belongs to H(div, Ω) = {q L 2 (Ω; R 2 ) : div q L 2 (Ω)} if and only if all the jumps [q R ] E := (q R + q R ) E, for E = + with ±, across an interior edge E disappear in their normal component [q R ] E ν E = 0 along E. Given an interior edge E E( ) shared by its neighbouring triangles + and and the vertices P ± opposite to E of ±, set { ± E 2 Ψ E (x) := (x P ± ±) for x ±, 0 elsewhere. A corresponding formula without applies to some boundary edge E E( ) with one neighbouring triangle +. hen Ψ E R 0 ( ) with supp Ψ E = w E := + defines an edge-basis function of R 0 ( ). Indeed, R 0 ( ) = span{ψ E : E E}. he Fortin interpolation operator I F q is defined for all q H 1 (Ω; R 2 ) by I F q = ( ) q ν E ds Ψ E E E E (with signs ± in the definition of ± and ν E = ν + ) such that (5.2) (q I F q) ν E = 0 along any E E. heorem 5.2. Let q H 1 (Ω; R 2 ) with Fortin interpolation I F q. hen it holds q I F q L 2 (Ω) C( ) h Dq L 2 (Ω). Proof. he condition (5.2) leads to the assumption of heorem 5.1 with q substituted by q I F q on any triangle. heorem 5.1 shows q I F q L 2 ( ) C(max )h D(q I F q) L 2 ( ). Any 2 2 matrix A with trace tr(a) = A 11 + A 22 and deviatoric part dev A = A tr(a)/2 I allows for the orthogonality of the 2 2 unit matrix I and dev A with respect to the scalar product A : B := j,k=1,2 A jkb jk of the two matrices A, B

14 14 C. CARSENSEN, J. GEDICKE, AND D. RIM R 2 2. he Pythagoras theorem shows for the associated Frobenius norm (i.e. A = A : A) A 2 = dev A 2 + tr(a) 2 /2. his identity for A = D(q I F q)(x) followed by an integration of x over leads to D(q I F q) 2 L 2 ( ) = dev D(q I F q) 2 L 2 ( ) + div(q I F q) 2 L 2 ( ) /2. Notice that DI F q = c I for some c R is constant on. Hence, dev DI F q = 0. Moreover, the Gauss divergence theorem and (5.2) show div q dx = q ν dx = (I F q) ν ds = div(i F q) dx = 2c. herefore, div(q I F q) has integral mean zero and so div(q I F q) L 2 ( ) div q L 2 ( ). Altogether, and with another application of the Pythagoras theorem, it follows D(q I F q) 2 L 2 ( ) dev Dq 2 L 2 ( ) + div q 2 L 2 ( ) /2 = Dq 2 L 2 ( ). he summation of the resulting estimate on q I F q 2 L 2 ( ) concludes the proof. over all 5.3. Raviart-homas MFEM. he mixed finite element method for the Poisson model problem of Subsection 4.1 with the Raviart-homas finite element space R 0 ( ) and the piecewise constant P 0 ( ) seeks (p R, u R ) R 0 ( ) P 0 ( ) with p R q R dx + u R div q R dx = 0 for all q R R 0 ( ); Ω Ω (5.3) v R div p R dx + fv R dx = 0 for all v R P 0 ( ). Let f (5.4) Ω Ω denote the piecewise L 2 projection of f onto P 0 ( ) with f := f := f(x) dx for all. heorem 5.3. here exists a unique solution (p R, u R ) of the Raviart-homas MFEM. he discrete flux p R is the unique minimiser of p q R L 2 (Ω) for all q R Q(f, ) := {q R R 0 ( ) : f + div q R = 0}.

15 EXPLICI ERROR ESIMAES FOR FEM 15 Proof. he existence of a unique solution follows from standard results in the theory of mixed FEM [BS08, Bra01, BF91]. he optimality is well known and follows from (5.3) for the test function q R := p R r R Q(0, ) for any r R Q(f, ). Indeed, (5.3) shows p R (p R r R ). Since p = u is a gradient, (p p R ) (p R r R ) and so p r R 2 L 2 (Ω) = p p R 2 L 2 (Ω) + p R r R 2 L 2 (Ω). he following immediate consequence of heorem 5.2 and 5.3 is announced as the a priori error estimate (1.3). Corollary 5.4. he Raviart-homas MFEM solution p R on Ω of the Poisson model problem (4.1) satisfies p p R L 2 (Ω) p I F p L 2 (Ω) C( ) h Dp L 2 (Ω). Proof. he first inequality follows from the minimising property of heorem 5.3 and the second from heorem 5.2. Remark 5.5 (Comparison with [MS09]). Corollary 5.4 is a significant improvement over [MS09]; the estimate [MS09, Equation (3.31)] is significantly greater than C(α) Numerical Example. able 5.1 displays the errors p p R L 2 (Ω), p I F p L 2 (Ω), and the efficiency index EI := (C( l ) h l Dp L 2 ( l ) )/ p p R L 2 (Ω) for different levels l in the benchmark problem from Subsection 4.4 based on the Matlab implementation [BC05]. l p p R L 2 (Ω) p I F p L 2 (Ω) EI able 5.1. Numerical results for Raviart-homas MFEM. 6. Crouzeix-Raviart NCFEM his section is devoted to the nonconforming finite element method (NCFEM) after Crouzeix and Raviart and its relation to the Raviart-homas MFEM usually associated with Marini [Mar85]. he implications lead to some equivalence of error estimates for the two methods.

16 16 C. CARSENSEN, J. GEDICKE, AND D. RIM 6.1. Crouzeix-Raviart NCFEM. he NCFEM after Crouzeix and Raviart concerns the nonconforming finite element space V NC ( ) := {v P 1 ( ) v continuous at mid(e), with v = 0 for mid( Ω E)}. he piecewise gradient NC : H 1 ( ) L 2 (Ω; R 2 ) is defined by ( NC v) := v for all and defines the scalar product a NC (u, v) := u v dx for all u, v H 1 ( ) and the induced discrete energy norm NC := a NC (, ). For every E E, the edge-oriented basis function ψ E is defined by ψ E (mid(e)) = 1 and ψ E (mid(f )) = 0 for all F E \ {E} and V NC ( ) = span{ψ E E E(Ω)}. he discrete Friedrichs inequality [BS08] reads v L 2 (Ω) C df v NC for all v V NC ( ). he constant C df does not depend on the mesh-size or cardinality of the shaperegular triangulation. he discrete Friedrichs inequality implies that NC is a norm on V NC ( ) and the Riesz representation theorem guarantees a unique solution u CR V NC of (6.1) a NC (u CR, v CR ) = fv CR dx for all v CR V NC ( ). Ω 6.2. Equivalence of CR-FEM and R-MFEM. he following equivalence theorem is well known [Mar85, BC05] and is given here to stress that the righthand side f in the Poisson model problem has to be modified to its piecewise integral mean f P 0 ( ) as in (5.4). For any set s 2 ( ) := E 2 = 36 mid( ) L 2 ( ) /. E E( ) he following theorem states a representation of the unique solution (5.3). heorem 6.1 (Marini [Mar85]). Suppose ũ CR V NC ( ) solves the discrete problem for the Crouzeix-Raviart FEM with modified right-hand side f P 0 ( ), i.e., ũ CR V NC ( ) satisfies (6.2) a NC (ũ CR, v CR ) = f v CR dx for all v CR V NC ( ). Ω

17 EXPLICI ERROR ESIMAES FOR FEM 17 hen the solution (p R, u R ) of (5.3) reads p R (x) = NC ũ CR f /2 (x mid( )) for x, u R = ũ CR dx + s 2 ( )f /144 on CR-FEM Error Estimate. his section establishes the error estimate (1.4) for the Crouzeix-Raviart nonconforming finite element method. he oscillations of a function f L 2 (Ω) are defined as osc(f, ) 2 := h 2 f f 2 L 2 ( ). heorem 6.2. he Crouzeix-Raviart NCFEM solution u CR V NC ( ) on Ω of the Poisson model problem (4.1) satisfies u u CR NC 1 1 osc(f, ) + + C( ) j 2 h D 2 u. L 2 (Ω) 1,1 Proof. Let ũ CR V NC ( ) solve (6.2) and set p CR := NC ũ CR P 0 ( ; R 2 ). he orthogonality of the L 2 projection Π 0 onto P 0 ( ) plus the Poincaré inequality show ũ CR u CR 2 NC = a NC (ũ CR, ũ CR u CR ) a NC (u CR, ũ CR u CR ) = (f f )(ũ CR u CR )dx Ω = (f f ) ( (ũ CR u CR ) Π 0 (ũ CR u CR ) ) dx Ω j 2 1,1 1 j 1,1 osc(f, ) ũ CR u CR NC. Hence, ũ CR u CR NC osc(f, )/j 1,1. he Pythagoras theorem leads to p p CR 2 L 2 (Ω) = p Π 0p 2 L 2 (Ω) + Π 0p p CR 2 L 2 (Ω). For the first term on the right-hand side, the Poincaré inequality yields p Π 0 p 2 L 2 (Ω) 1 h j1,1 2 Dp 2 L 2 (Ω). For the second term, heorem 6.1 leads to Π 0 p R = p CR and therefore Π 0 p p CR 2 L 2 (Ω) = Π 0(p p R ) 2 L 2 (Ω) p p R 2 L 2 (Ω). hus, Corollary 5.4 and the triangle inequality conclude the proof.

18 18 C. CARSENSEN, J. GEDICKE, AND D. RIM Remark 6.3. he estimate in [MS09, heorem 4.1] is wrong in the sense that the difference of u CR and ũ CR has been neglected. Even the corrected version of that estimate is less sharp than heorem 6.2 because of Remark Numerical Example. able 6.1 displays the error u u CR NC, the oscillations osc(f, ), and the efficiency index ( 1 1 EI := osc(f, ) + + C( ) j 1,1 j1,1 2 2 h D 2 u ) L / u u 2 (Ω) CR NC in the benchmark problem from Subsection 4.4. l u u CR NC osc(f, ) EI able 6.1. Numerical results of Crouzeix-Raviart FEM. Acknowledgements he authors would like to thank the two anonymous referees for their valuable comments and suggestions. In particular, to the second referee which pointed us to the recently improved Poincaré constant for triangles. his research was supported by the World Class University (WCU) program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and echnology R and the DFG Research Center MAHEON Mathematics for key technologies in Berlin. he second author was supported in parts by the DFG graduate school BMS Berlin Mathematical School. References [AB85] D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, [ACF99] J. Alberty, C. Carstensen, and S. A. Funken, Remarks around 50 lines of Matlab: short finite element implementation, Numer. Algorithms 20 (1999), no. 2-3, 117 [AD99] 137. G. Acosta and R. G. Durán, he maximum angle condition for mixed and nonconforming elements: application to the Stokes equations, SIAM J. Numer. Anal. 37 (1999), no. 1, [Arb82] P. Arbenz, Computable finite element error bounds for Poisson s equation, IMA J. Numer. Anal. 2 (1982), no. 4,

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20 20 C. CARSENSEN, J. GEDICKE, AND D. RIM Carsten Carstensen, Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, Berlin, Germany, Department of Computational Science and Engineering, Yonsei University, Seoul, Korea address: Joscha Gedicke, Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, Berlin, Germany address: Donsub Rim, Yonsei School of Business and Department of Computational Science and Engineering, Yonsei University, Seoul, Korea address: