NUMERICAL PARTIAL DIFFERENTIAL EQUATIONS

Transcription

1 NUMERICAL PARTIAL DIFFERENTIAL EQUATIONS Lecture Notes, Winter 2011/12 Christian Clason January 31, 2012 Institute for Mathematics and Scientific Computing Karl-Franzens-Universität Graz

2 CONTENTS I BACKGROUND 1 overview of the finite element method Variational form of PDEs Ritz Galerkin approximation Approximation by piecewise polynomials Implementation A posteriori error estimates and adaptivity 10 2 variational theory of pdes Function spaces Weak solutions of elliptic PDEs 18 II CONFORMING FINITE ELEMENTS 3 galerkin approach for variational problems 25 4 finite element spaces Construction of finite element spaces Examples of finite elements The interpolant 34 5 polynomial interpolation in sobolev spaces The Bramble Hilbert lemma Interpolation error estimates 40 6 implementation Triangulation Assembly 45 i

3 contents 7 error estimates for the finite element approximation A priori error estimates A posteriori error estimates 50 III NONCONFORMING FINITE ELEMENTS 8 generalized galerkin approach 56 9 discontinuous galerkin methods Weak formulation of advection-reaction equations Galerkin approach Error estimates mixed methods Abstract saddle point problems Galerkin approximation of saddle point problems A mixed method for the Poisson equation 77 IV TIME-DEPENDENT PROBLEMS 11 variational theory of parabolic pdes Function spaces Weak solution of parabolic PDEs galerkin approach for parabolic problems Time stepping methods Discontinuous Galerkin methods for parabolic problems A priori error estimates 92 ii

4 Part I BACKGROUND

5 INTRODUCTION Partial differential equations appear in many mathematical models of physical, biological and economic phenomena, such as elasticity, electromagnetics, fluid dynamics, quantum mechanics, pattern formation or derivative valuation. However, closed-form or analytic solutions of these equations are only available in very specific cases (e.g., for simple geometries or constant coefficients), and so one has to resort to numerical approximations of these solutions. In these notes, we will consider finite element methods, which have developed into one of the most flexible and powerful frameworks for the numerical (approximate) solution of partial differential equations. They were first proposed by Richard Courant [Courant 1943]; but the method did not catch on until engineers started applying similar ideas in the early 1950s. Their mathematical analysis began later, with the works of Miloš Zlámal [Zlámal 1968]. Knowledge of real analysis (in particular, Lebesgue integration theory) and functional analysis (especially Hilbert space theory) as well as some familiarity of the weak theory of partial differential equations is assumed, although the fundamental results of the latter (Sobolev spaces and the variational formulation of elliptic equations) are recalled in Chapter 2. These notes are mostly based on the following works: [1] E. Süli (2011). Finite Element Methods for Partial Differential Equations. Lecture notes. url: [2] R. Rannacher (2008). Numerische Mathematik 2. Lecture notes. url: iwr.uni-heidelberg.de/~lehre/notes/num2/numerik2.pdf [3] S. C. Brenner and L. R. Scott (2008). The Mathematical Theory of Finite Element Methods. 3rd ed. Vol. 15. Texts in Applied Mathematics. Springer, New York [4] D. Braess (2007). Finite Elements. 3rd ed. Cambridge University Press, Cambridge [5] A. Ern and J.-L. Guermond (2004). Theory and Practice of Finite Elements. Vol Applied Mathematical Sciences. Springer-Verlag, New York [6] V. Thomée (2006). Galerkin Finite Element Methods for Parabolic Problems. 2nd ed. Vol. 25. Springer Series in Computational Mathematics. Springer-Verlag, Berlin 2

6 OVERVIEW OF THE FINITE ELEMENT METHOD We begin with a bird s-eye view of the finite element method by considering a simple onedimensional example. Since the goal here is to give the flavor of the results and techniques used in the construction and analysis of finite element methods, not all arguments will be completely rigorous (especially those involving derivatives and function spaces). These gaps will be filled by the more general theory in the following chapters variational form of pdes Consider for a given function f the two-point boundary value problem (BVP) { u (x) = f(x) for x (0, 1), u(0) = 0, u (1) = 1. Similar to Krylov methods, the idea is to pass from this differential equation to a system of linear equations which can be solved on a computer by projection onto a finite-dimensional subspace. Any projection requires a kind of inner product, which we introduce now. We begin by multiplying this equation with any sufficiently regular function v with v(0) = 0, integrating over x [0, 1] and integrating by parts. Then any solution u of (BVP) satisfies (f, v) := = f(x)v(x) dx = u (x)v (x) dx =: a(u, v), 1 0 u (x)v(x) dx where we have used that u (1) = 0 and v(0) = 0. Let us (formally for now) define the space V := { v L 2 (0, 1) : a(v, v) <, v(0) = 0 }. 3

7 1 overview of the finite element method Then we can pose the following problem: Find u V such that (W) a(u, v) = (f, v) for all v V holds. This is called the weak or variational form of (BVP) (since v varies over all V). If the solution u of (W) is twice continuously differentiable and f is continuous, one can prove (by taking suitable test functions v) that u satisfies (BVP). On the other hand, there are solutions of (W) even for discontinuous f L 2 (0, 1). Since then the second derivative of u is discontinuous, u cannot be a solution of (BVP). For this reason, u V satisfying (W) is called a weak solution of (BVP). Note that the Dirichlet boundary condition u(0) = 0 appears explicitly in the definition of V, while the Neumann condition u (1) = 0 is implicitly incorporated in the variational formulation. In the context of finite element methods, Dirichlet conditions are therefore frequently called essential conditions, while Neumann conditions are referred to as natural conditions. 1.2 ritz galerkin approximation The fundamental idea is now to approximate u by considering (W) on a finite-dimensional subspace S V. We are thus looking for u S S satisfying (W S ) a(u S, v) = (f, v) for all v S. Note that this is still the same equation; only the function spaces have changed. This is a crucial point in (conforming) finite element methods. (Nonconforming methods, for which S V or v / V, will be treated in Part C.) Since S is finite-dimensional, there exists a basis ϕ 1,..., ϕ n of S. Due to the bilinearity of a(, ), it suffices to require u S = n i=1 U iϕ i S to satisfy If we define a(u S, ϕ j ) = (f, ϕ j ) for all 1 j n. U = (U 1,..., U n ) T R n, F = (F 1,..., F n ) T R n, F i = (f, ϕ i ), K = (K ij ) R n n, K ij = a(ϕ i, ϕ j ), we have that u S satisfies (W S ) if and only if ( iff ) KU = F. This linear system has a unique solution iff KV = 0 implies V = 0. To show this, we set v := n i=1 V iϕ i S. Then, 0 = KV = (a(v, ϕ 1 ),..., a(v, ϕ n )) T 4

8 1 overview of the finite element method implies that 0 = n V i a(v, ϕ i ) = a(v, v) = i=1 1 0 v (x) 2 dx. This means that v must vanish almost everywhere and thus v is constant. (This argument will be made rigorous in the next chapter.) Since v(0) = 0, we deduce v 0, and hence, by the linear independence of the ϕ, V i = 0 for all 1 i n. There are two remarks to made here. First, we have argued unique solvability of the finitedimensional system by appealing to the properties of the variational problem to be approximated. This is a standard argument in finite element methods, and the fact that the approximation inherits the well-posedness of the variational problem is one of the strengths of the Galerkin approach. Second, this argument shows that the stiffness matrix K is (symmetric and) positive definite, since V T KV = a(v, v) > 0 for all V 0. Now that we have an approximate solution u S S, we are interested in estimating the discretization error u S u, which of course depends on the choice of S. The fundamental observation is that by subtracting (W) and (W S ) for the same test function w S, we obtain a(u u S, w) = 0 for all w S. This key property is called Galerkin orthogonality, and expresses that the discretization error is (in some sense) orthogonal to S. This can be exploited to derive error estimates in the energy norm v 2 E = a(v, v) for v V. It is straightforward to verify that this indeed defines a norm, which satisfies the Cauchy Schwarz inequality We can thus show that for any v S, a(v, w) v E w E for all v, w V. u u S 2 E = a(u u S, u v) + a(u u S, v u S ) = a(u u S, u v) u u S E u v E due to the Galerkin orthogonality for v u S S. Taking the infimum over all v, we obtain u u S E inf v S u v E, and equality holds and hence this infimum is attained for u S S solving (W S ). The discretization error is thus completely determined by the approximation error of the solution u of (W) by functions in S: (1.1) u u S E = min v S u v E. 5

9 1 overview of the finite element method To derive error estimates in the L 2 (0, 1) norm 1 v 2 L = (v, v) = v(x) 2 dx, 2 0 we apply a duality argument (also called Aubin Nitsche trick). Let w be the solution of the dual (or adjoint) problem (1.2) { w (x) = u(x) u S (x) for x (0, 1), w(0) = 0, w (1) = 1. Inserting this into the error and integrating by parts (using (u u S )(0) = w (1) = 0), we obtain for all v S the estimate u u S 2 L 2 = (u u S, u u S ) = (u u S, w ) = (u, w ) (u S, w ) = a(u u S, w) a(u u S, v) = a(u u S, w v) u u S E w v E. Dividing by u u S L 2 yields = w L 2, inserting (1.2) and taking the infimum over all v S u u S L 2 inf v S w v E u u S E w 1 L 2. To continue, we require an approximation property for S: There exists ε > 0 such that (1.3) inf v S f v E ε f L 2 holds for sufficiently smooth f V. If we can apply this estimate to w and u, we obtain u u S L 2 ε u u S E = ε min v S u v E ε 2 u L 2 = ε 2 f L 2. This is another key observation: The error estimate depends on the regularity of the weak solution u, and hence on the data f. The smoother u, the better the approximation. The finite element method is characterized by a special class of subspaces of piecewise polynomials which have these approximation properties. 1.3 approximation by piecewise polynomials Given a set of nodes 0 = x 0 < x 1 < < x n = 1, 6

10 1 overview of the finite element method set S = { v C 0 (0, 1) : v [xi 1,x i ] P 1 and v(0) = 0 }, where P 1 is the space of all linear polynomials. (The fact that S V is not obvious, and will be proved later.) This is a subspace of the space of linear splines. A basis of S, which is especially convenient for the implementation, is formed by the linear B-splines (hat functions) x x i 1 x i x i 1 if x [x i 1, x i ], x ϕ i (x) = i+1 x x i+1 x i if x [x i, x i+1 ], 0 else, for 1 i n, which satisfy ϕ i (x j ) = δ ij := { 1 if i = j, 0 if i j. This nodal basis property immediately yields linear independence of the ϕ i. To show that the ϕ i span S, we consider the interpolant of v V, defined as v I := n v(x i )ϕ i (x) S. i=1 For v S, the interpolation error v v I is piecewise linear as well, and since (v v I )(0) = 0, this implies that v v I 0. Any v S can thus be written as a linear combination of ϕ i (given by its interpolant), and hence the ϕ i form a basis of S. We also note that this implies that the interpolation operator I : V S, v v I is a projection. We are now in a position to prove the approximation property of S. Let h := max 1 i n h i, h i := (x i x i 1 ) denote the mesh size. We wish to show that there exists a constant C > 0 such that for all sufficiently smooth u V, u u I E Ch u L 2. It suffices to consider this error separately on each element [x i 1, x i ], i.e., to show xi x i 1 (u u I ) (x) 2 dx C 2 h 2 i xi x i 1 u (x) 2 dx. Furthermore, since u I is piecewise linear, the error e := u u I satisfies (e [xi 1,x i ]) = (u [xi 1,x i ]). Using the affine transformation ẽ(t) := e(x(t)) with x(t) = x i 1 + t(x i x i 1 ) (a scaling argument), we just need to prove (1.4) 1 ẽ (t) 2 dt c ẽ (t) 2 dt. 7

11 1 overview of the finite element method (This is an elementary version of Poincaré s inequality). Since u I is the nodal interpolant of u, the error satisfies ẽ(x i 1 ) = ẽ(x i ) = 0. In addition, u I is linear and u continuously differentiable on [x i 1, x i ]. By Rolle s theorem, there hence exists a ξ (0, 1) with ẽ (ξ) = 0. Thus, for all y [0, 1] we have (with b a f(t) dt = a b f(t) dt for a > b) ẽ (y) = y ξ w (t) dt. We can now use the Cauchy Schwarz inequality to estimate y 2 y ẽ (y) 2 = ẽ (t) 1 dt y ẽ (t) 2 dt ξ y ξ 1 0 ξ ẽ (t) 2 dt. Integrating both sides with respect to y and taking the supremum over all ξ (0, 1) yields (1.4) with c := 1 sup ξ (0,1) 0 ξ y ξ dy = 1 2. Summing over all elements and estimating h i by h shows the approximation property (1.3) for S with ε := ch. For this choice of S, the solution u S of (W S ) satisfies (1.5) u u S L 2 c 2 h 2 u L 2. This is called an a priori estimate, since it only requires knowledge of the given data f = u, but not of the solution u S. It tells us that if we can make the mesh size h arbitrarily small, we can approximate the solution u of (W) arbitrarily well. Note that the power of h is one order higher for the L 2 (0, 1) norm compared to the energy norm. 1.4 implementation As seen in section 1.2, the numerical computation of u S S boils down to solving the linear system KU = F for the vector of coefficients U, e.g., by the method of conjugate gradients (since K is symmetric and positive definite). The missing step is the computation of the elements K ij = a(ϕ i, ϕ j ) of K and the entries F j = (f, ϕ j ) of F. (This procedure is called assembly.) In principle, this can be performed by computing the integrals for each pair (i, j) in a nested loop (node-based assembly). A more efficient approach (especially in higher dimensions) is element-based assembly: The integrals are split into sums of contributions from each element, e.g., a(ϕ i, ϕ j ) = 1 0 ϕ i(x)ϕ j(x) dx = n k=1 xk x k 1 ϕ i(x)ϕ j(x) dx, 8

12 1 overview of the finite element method and the contributions from a single element for all (i, j) are computed simultaneously. Here we can exploit that by the definition, ϕ i is non-zero only on the two elements [x i 1, x i ] and [x i, x i+1 ]. Hence, for each element [x k 1, x k ], the integrals are non-zero only for pairs (i, j) with k 1 i, j k. Note that this implies that K is tridiagonal and therefore sparse (meaning that the number of non-zero elements grows as n, not n 2 ), which allows efficient solution of the linear system even for large n. Another useful observation is that except for an affine transformation, the basis functions are the same on each element. We can thus use the substitution rule to transform the integrals over [x k 1, x k ] to the reference element [0, 1]. Setting ξ(x) = x x k 1 x k x k 1 and ^ϕ 1 (ξ) = 1 ξ, ^ϕ 2 (ξ) = ξ, we have that ϕ k 1 (x) = ^ϕ 1 (ξ(x)) and ϕ k (x) = ^ϕ 2 (ξ(x)) and thus that where xk x k 1 ϕ i(x)ϕ j(x) dx = (x k x k 1 ) 1 τ(i) = 1 0 { 1 if i = k 1 2 if i = k, ^ϕ τ(i)(ξ) ^ϕ τ(j)(ξ) dξ, is the so-called global-to-local index. (Correspondingly, the inverse mapping τ 1 is called the local-to-global index.) The contribution from the element [x k 1, x k ] to a(ϕ i, ϕ j ) is thus { h 1 k if i = j, a k (ϕ i, ϕ j ) = h 1 k if i j. The right-hand side (f, ϕ j ) can be computed in a similar way, using numerical quadrature if necessary. Alternatively, one can replace f by its nodal interpolant f I = n i=0 f(x i)ϕ i and use n (f, ϕ j ) (f I, ϕ j ) = f(x i ) (ϕ i, ϕ j ) =: Mf. i=0 The elements M ij := (ϕ i, ϕ j ) of the mass matrix M are again computed elementwise using transformation to the reference element: xk { 1 hk if i = j, 3 ϕ i (x)ϕ j (x) dx = h k ^ϕ τ(i) (ξ) ^ϕ τ(j) (ξ) dξ = h k x k if i j. This can be done at the same time as assembling K. Finally, the Dirichlet condition u(0) = 0 is enforced by replacing the first equation in the linear system by U 0 = 0, i.e., replacing the first row of K by (1, 0,... ) T and the first element of Mf by 0. The main advantage is that this procedure can easily be extended to non-homogeneous Dirichlet conditions. The full algorithm (in matlab-like notation) for our boundary value problem is given in Algorithm

13 1 overview of the finite element method Algorithmus 1.1 Finite element method in 1d Input: 0 = x 0 < < x n = 1, F := (f(x 0 ),..., f(x n )) T 1: Set K ij = M ij = 0 2: for k = 1,..., n do 3: Set h k = x k x k 1 ( ) 1 1 4: Set K k 1:k,k 1:k K k 1:k,k 1:k + 1 h k 1 1 ( ) 5: Set M k 1:k,k 1:k M k 1:k,k 1:k + h k : end for 7: K 0,1:n = 0, K 0,0 = 1, M 0,0:n = 0 8: Solve KU = MF Output: U a posteriori error estimates and adaptivity The a priori estimate (1.5) is important for proving convergence as the mesh size h 0, but often pessimistic in practice since it depends on the global regularity of u. If u (x) is large only in some parts of the domain, it would be preferable to reduce the mesh size locally. For this, a posteriori estimates are useful, which involve the computed solution u S but are able to give information on which elements should be refined (i.e., replaced by a larger number of smaller elements). We consider again the space S of piecewise linear finite elements on the nodes x 0,..., x N with mesh size h, as defined in section 1.3. We once more apply a duality trick: Let w be the solution of { w (x) = u(x) u S (x) for x (0, 1), and proceed as before, yielding w(0) = 0, w (1) = 1, u u S 2 L 2 = a(u u S, w v) for all v S. We now choose v = w I S, the interpolant of w. Then we have u u S 2 L 2 = a(u u S, w w I ) = a(u, w w I ) a(u S, w w I ) = (f, w w I ) a(u S, w w I ). Note that the unknown solution u of (W) no longer appears on the right hand side. We now use the specific choice of v to localize the error inside each element [x i 1, x i ]: Writing the integrals over [0, 1] as sums of integrals over the elements, we can integrate by parts on each 10

14 1 overview of the finite element method element and use the fact that (w w I )(x i ) = 0 to obtain u u S 2 L 2 = n = xi i=1 xi n i=1 ( xi n i=1 x i 1 f(x)(w w I )(x) dx n i=1 x i 1 (f + u S)(x)(w w I )(x) dx x i 1 (f + u S)(x) 2 dx ) 1 2 ( xi xi x i 1 u S(x)(w w I ) (x) dx x i 1 (w w I )(x) 2 dx by the Cauchy Schwarz inequality. The first term contains the finite element residual R h := f + u S, which we can evaluate after computing u S. For the second term, one can show (similarly as in the proof of the inequality (1.5)) that ) 1 2 ( xi holds, from which we deduce ) 1 2 (w w I )(x) 2 dx w wi L2 h2 i x i 1 4 w L 2 u u S 2 L w L 2 n h 2 i R h L 2 (x i 1,x i ) i=1 = 1 4 u u S L 2 n h 2 i R h L 2 (x i 1,x i ) i=1 by the definition of w. This yields the a posteriori estimate u u S L n h 2 i R h L 2 (x i 1,x i ). i=1 This estimate can be used for an adaptive procedure: Given a tolerance τ > 0, 1: Choose initial mesh 0 < x (0) 0 <... x (0) = 1, compute corresponding solution u N (0) S (0), evaluate R h (0), set m = 0 2: while n i=1 (h(m) i ) 2 R h (m) < τ do L 2 (x (m) i 1,x(m) i ) 3: Choose new mesh 0 < x (m+1) 0 <... x (m+1) = 1 N (m+1) 4: compute corresponding solution u S (m+1) 5: evaluate R h (m+1) 6: set m m + 1 7: end while 11

15 1 overview of the finite element method There are different strategies to choose the new mesh. They should be reliable, meaning that the error on the new mesh in a certain norm can be guaranteed to be less than a given tolerance, and efficient, meaning that the number of new nodes should not be larger than necessary. One (simple) possibility is to refine those elements where R h is largest (or larger than a given threshold) by replacing them with two elements of half size. 12

16 VARIATIONAL THEORY OF PDES In this chapter, we collect for the most part without proof some necessary results from functional analysis and the weak theory of (elliptic) partial differential equations. Details and proofs can be found in, e.g., [Adams and Fournier 2003], [Evans 2010] and [Zeidler 1995a] function spaces As we have seen, the regularity of the solution of partial differential equations plays a crucial role in how well it can be approximated numerically. This regularity can be described by the two properties of (Lebesgue-)integrability and differentiability. lebesgue spaces, Let Ω be an open subset of R n, n N 0. We recall that for 1 p } L p (Ω) := {f measurable : f L p(ω) < with f L p (Ω) = ( f L (Ω) Ω f(x) p dx) 1 p for 1 p <, = ess sup f(x) x Ω are Banach spaces of (equivalence classes up to equality apart from a set of zero measure of) Lebesgue-integrable functions. The corresponding norms satisfy Hölder s inequality fg L 1 (Ω) f L p (Ω) f L q (Ω) if p 1 + q 1 = 1 (with 1 := 0). For bounded Ω, this implies (by using g 1) that L p (Ω) L q (Ω) for p q. We will also use the space L 1 loc(ω) := { f : f K L 1 (K) for all compact K Ω }. 13

17 2 variational theory of pdes For p = 2, L p (Ω) is a Hilbert space with inner product (f, g) := f, g L 2 (Ω) = f(x)g(x) dx, and Hölder s inequality for p = q = 2 reduces to the Cauchy Schwarz inequality. Ω hölder spaces We now consider functions which are continuously differentiable. It will be convenient to use a multi-index α := (α 1,..., α n ) N n, for which we define its length α := n i=1 α i, to describe the (partial) derivative of order α D α f(x 1,..., x n ) := α f(x 1,..., x n ) x α 1 1 xα n n For brevity, we will often write i := x i. We denote by C k (Ω) the set of all continuous functions f for which D α f is continuous for all α k. If Ω is bounded, C k (Ω) is the set of all functions in C k (Ω) for which all D α f can be extended to a continous function on Ω, the closure of Ω. These spaces are Banach spaces if equipped with the norm f C k (Ω) = α k x Ω sup D α f(x). Finally, we define C k 0 (Ω) as the space of all f Ck (Ω) whose support (the closure of {x Ω : f(x) 0}) is a bounded subset of Ω, as well as C 0 (Ω) = k 0 C k 0(Ω) (and similarly C (Ω)). sobolev spaces If we are interested in weak solutions, it is clear that the Hölder spaces entail a too strong notion of (pointwise) differentiability. All we required is that the derivative is integrable, and that an integration by parts is meaningful. This motivates the following definition: A function f L 1 loc (Ω) has a weak derivative if there exists g L1 loc (Ω) such that (2.1) g(x)ϕ(x) dx = ( 1) α f(x)d α ϕ(x) dx Ω for all ϕ C 0 (Ω). In this case, the weak derivative is (uniquely) defined as Dα f := g. For f C k (Ω), the weak derivative coincides with the usual (pointwise) derivative (justifying the abuse of notation), but the weak derivative exists for a larger class of functions such as Ω 14

18 2 variational theory of pdes continuous and piecewise smooth functions. For example, f(x) = x, x Ω = ( 1, 1), has the weak derivative Df(x) = sign(x), while Df(x) itself does not have any weak derivative. We can now define the Sobolev spaces W k,p (Ω) for k N 0 and 1 p : W k,p (Ω) = {f L p (Ω) : D α f L p (Ω) for all α k}, which are Banach spaces when endowed with the norm f W k,p (Ω) = D α f p L p (Ω) f W k, (Ω) = α k α k D α f L (Ω). We shall also use the corresponding semi-norms f W k,p (Ω) = D α f p L p (Ω) f W k, (Ω) = α =k α =k D α f L (Ω). 1 p 1 p for 1 p <, for 1 p <, We are now concerned with the relation between the different norms introduced so far. For many of these results to hold, we require that the boundary Ω of Ω is sufficiently smooth. We shall henceforth assume that Ω R n has a Lipschitz boundary, meaning that Ω can be parametrized by a finite set of functions which are uniformly Lipschitz continuous. (This condition is satisfied, for example, by polygons for n = 2 and polyhedra for n = 3.) A fundamental result is then the following approximation property (which does not hold for arbitrary domains). Theorem 2.1 (Density 1 ). For 1 p < and any k N 0, C (Ω) is dense in W k,p (Ω). This theorem allows us to prove results for Sobolev spaces such as chain rules by showing them for smooth functions (in effect, transferring results for usual derivatives to their weak counterparts). This is called a density argument. The next theorem states that, within limits determined by the spatial dimension, we can trade differentiability for integrability for Sobolev space functions. 1 Originally shown in a paper by Meyers and Serrin rightfully celebrated both for its content and the brevity of its title, H = W. For the proof, see, e.g., [Evans 2010, 5.3.3, Th. 3], [Adams and Fournier 2003, Th. 3.17] 15

19 2 variational theory of pdes Theorem 2.2 (Sobolev 2, Rellich Kondrachov 3 embedding). Let 1 p, q < and Ω R n be a bounded open set with Lipschitz boundary. Then, the following embeddings are continuous: L q (Ω) if p < n np and p q, k n p W k,p (Ω) L q (Ω) if p = n and p q <, k C 0 (Ω) if p > n. k Moreover, the following embeddings are compact: { L q W k,p (Ω) if p n n pk and 1 q < k np (Ω), C 0 (Ω) if p > n. k In particular, the embedding W k,p (Ω) W k 1,p (Ω) is compact for all k and 1 p. We now prove a generalization of the observation that piecewise linear and continuous functions have a weak derivative. Theorem 2.3. Let Ω R n be a bounded Lipschitz domain which can be partitioned into N N Lipschitz subdomains Ω j, i.e., Ω = N j=1 Ω j. Then, for every k 1 and 1 p, { v C k 1 (Ω) : v Ωj C k (Ω j ), 1 j N } W k,p (Ω). Proof. It suffices to show the inclusion for k = 1. Let v C 0 (Ω) such that v Ωj C 1 (Ω j ) for all 1 j N. We need to show that i v exists as a weak derivative for all 1 i n and that i v L p (Ω). An obvious candidate is { i v Ωj (x) if x Ω j, 1 j N, w i := c else for arbitrary c R. By the embedding C 0 (Ω j ) L (Ω j ) and the boundedness of Ω, w i L p (Ω) for any 1 p. It remains to verify (2.1). By splitting the integration into a sum over the Ω j and integrating by parts on each subdomain (where v is continuously differentiable), we obtain for any ϕ C 0 (Ω) Ω N w i ϕ dx = i v Ωi ϕ dx j=1 Ω j N N = v Ωj ϕ (ν j ) i dx v Ωj i ϕ dx j=1 Ω j j=1 Ω j N = v Ωj ϕ (ν j ) i dx v i ϕ dx, Ω j Ω j=1 2 e.g., [Evans 2010, 5.6], [Adams and Fournier 2003, Th. 4.12] 3 e.g., [Evans 2010, 5.7], [Adams and Fournier 2003, Th. 6.3] 16

20 2 variational theory of pdes where ν j = ((ν j ) 1,..., (ν j ) n ) is the outer normal vector on Ω j, which exists almost everywhere since Ω j is a Lipschitz domain. Now the sum over the boundary integrals vanishes since either ϕ(x) = 0 if x Ω j Ω or v Ωj (x)ϕ(x)(ν j ) i (x) = v Ωk (x)ϕ(x)(ν k ) i (x) if x Ω j Ω k due to the continuity of v. This implies i v = w i by definition. Next, we would like to see how Dirichlet boundary conditions make sense for weak solutions. For this, we define a trace operator T (via limits of continous functions) which maps a function f on a bounded domain Ω R n to a function Tf on Ω. Theorem 2.4 (Trace theorem 4 ). Let kp < n and q (n 1)p/(n kp), and Ω R n be a bounded open set with Lipschitz boundary. Then, T : W k,p (Ω) L q ( Ω) is a bounded linear operator, and there exists a constant C > 0 depending only on p and Ω such that for all f W k,p (Ω), If kp = n, this holds for any p q <. This implies (although it is not obvious 5 ) that Tf L q ( Ω) C f W k,p (Ω). W k,p 0 (Ω) := { f W k,p (Ω) : T(D α f) = 0 L p ( Ω) for all α < k } is well-defined, and that W k,p (Ω) C 0 (Ω) is dense in Wk,p 0 (Ω). For functions in W 1,p 0 (Ω),the semi-norm W 1,p (Ω) is equivalent to the full norm W 1,p (Ω). Theorem 2.5 (Poincaré s inequality 6 ). Let 1 p < and let Ω be a bounded open set. Then, there exists a constant C > 0 depending only on p and Ω such that for all f W 1,p 0 (Ω), holds. f W 1,p (Ω) C f W 1,p (Ω) The proof is very similar to the argumentation in Chapter 1, using the density of C 0 (Ω) in W 1,p 0 (Ω); in particular, it is sufficient that Tf is zero on a part of the boundary Ω of non-zero measure. In general, we have that any f W 1,p (Ω), 1 p, for which D α f = 0 almost everywhere in Ω for all α = 1 must be constant. Again, W k,p (Ω) is a Hilbert space for p = 2, with inner product f, g W k,2 (Ω) = α k (D α f, D α g). 4 e.g., [Evans 2010, 5.5], [Adams and Fournier 2003, Th. 5.36] 5 e.g., [Evans 2010, 5.5, Th. 2], [Adams and Fournier 2003, Th. 5.37] 6 e.g, [Adams and Fournier 2003, Cor. 6.31] 17

21 2 variational theory of pdes For this reason, one usually writes H k (Ω) := W k,2 (Ω). In particular, we will often consider H 1 (Ω) := W 1,2 (Ω) and H 1 0 (Ω) := W1,2 0 (Ω). With the usual notation f := ( 1f,..., n f) for the gradient of f, we can write f H 1 (Ω) = f L 2 (Ω) for the semi-norm on H 1 (Ω) (which, by the Poincaré inequality 2.5, is a full norm on H 1 0 (Ω)) and f, g H 1 (Ω) = (f, g) + ( f, g) for the inner product on H 1 (Ω). Finally, we denote the topological dual of H 1 0 (Ω) (i.e., the space of all continuous linear functionals on H 1 0 (Ω)) by H 1 (Ω) := (H 1 0 (Ω)), which is endowed with the operator norm f H 1 (Ω) = sup f, ϕ H 1(Ω),H1 ϕ H 1 0 (Ω),ϕ 0 ϕ H 1 0 (Ω) where f, ϕ V,V := f(ϕ) denotes the duality pairing between a Banach space V and its dual V. We can now tie together some loose ends from Chapter 1. The space V can now be rigorously defined as V := { v H 1 (0, 1) : v(0) = 0 }, which makes sense due to the embedding (for n = 1) of H 1 (0, 1) in C([0, 1]). Due to Poincaré s inequality, v 2 = a(v, v) = 0 implies v H 1 (Ω) H 1 (Ω) = 0 and hence v = 0. Similarly, the existence of a unique weak solution u V follows from the Riesz representation theorem. Finally, Theorem 2.3 guarantees that S V. 0 (Ω), 2.2 weak solutions of elliptic pdes In most of these notes, we consider boundary value problems of the form (2.2) n j,k=1 j (a jk (x) k u) + n b j (x) j u + c(x)u = f j=1 on a bounded open set Ω R n, where a jk, b j, c and f are given functions on Ω. We do not fix boundary conditions at this time. This problem is called elliptic if there exists a constant α > 0 such that (2.3) n j,k=1 a jk (x)ξ j ξ k α n ξ 2 j for all ξ R n, x Ω. j=1 18

22 2 variational theory of pdes Assuming all functions and the domain are sufficiently smooth, we can multiply by a smooth function v, integrate over x Ω and integrate by parts to obtain (2.4) n n n (a jk j u, k v) + (b j j u, v) + (cu, v) + (a jk k uν j, v) Ω = (f, v), j,k=1 j=1 j,k=1 where ν := (ν 1,..., ν n ) T is the outward unit normal on Ω and (f, g) Ω := f(x)g(x) dx. Ω Note that this formulation only requires a jk, b j, c L (Ω) and f L 2 (Ω) in order to be well-defined. We then search for u V satisfying (2.4) for all v V including boundary conditions which we will discuss next. We will consider the following three conditions: dirichlet conditions We require u = g on Ω (in the sense of traces) for given g L 2 ( Ω). If g = 0 (homogeneous Dirichlet conditions), we take V = H 1 0 (Ω), in which case the boundary integrals in (2.4) vanish since v = 0 on Ω. The weak formulation is thus: Find u H 1 0 (Ω) satisfying for all v H 1 0 (Ω). a(u, v) := n j,k=1 (a jk j u, k v) + n (b j j u, v) + (cu, v) = (f, v) If g 0, and g and Ω are sufficiently smooth (e.g., g H 1 ( Ω) with Ω of class C 1 ) 7, we can find a function u g H 1 (Ω) such that Tu g = g. We then set u = ũ + u g, where ũ H 1 0 (Ω) satisfies for all v H 1 0 (Ω). j=1 a(u, v) = (f, v) a(u g, v) neumann conditions We require n j,k=1 a jk k uν j = g on Ω for given g L 2 ( Ω). In this case, we can substitute this equality in the boundary integral in (2.4) and take V = H 1 (Ω). We then look for u H 1 (Ω) satisfying a(u, v) = (f, v) + (g, v) Ω for all v H 1 (Ω), where v in the last inner product should be understood in the sense of traces, i.e., as Tv. 7 [Renardy and Rogers 2004, Th. 7.40] 19

23 2 variational theory of pdes robin conditions We require du+ n j,k=1 a jk k uν j = g on Ω for given g L 2 ( Ω) and d L ( Ω). Again we can substitute this in the boundary integral and take V = H 1 (Ω). The weak form is then: Find u H 1 (Ω) satisfying for all v H 1 (Ω). a R (u, v) := a(u, v) + (du, v) Ω = (f, v) + (g, v) Ω These problems have a common form: For a given Hilbert space V, a bilinear form a : V V R and a linear functional F : V R (e.g., F : v (f, v) in the case of Dirichlet conditions), find u V such that (2.5) a(u, v) = F(v), for all v V. The existence and uniqueness of a solution can be guaranteed by the Lax Milgram theorem, which is a generalization of the Riesz representation theorem (note that a is in general not symmetric). Theorem 2.6 (Lax Milgram theorem). Let a Hilbert space V, a bilinear form a : V V R and a linear functional F : V R be given satisfying the following conditions: (i) Coercivity: There exists a c 1 > 0 such that a(v, v) c 1 v 2 V for all v V. (ii) Continuity: There exists c 2, c 3 > 0 such that a(v, w) c 2 v V w V, F(v) c 3 v V for all v, w V. Then, there exists a unique solution u V to problem (2.5) satisfying (2.6) u V 1 c 1 F V. Proof. For every fixed u V, the mapping v a(u, v) is a linear functional on V, which is continuous by assumption (ii), and so is F. By the Riesz Fréchet representation theorem, 8 there exist unique ϕ u, ϕ F V such that ϕ u, v V = a(u, v) and ϕ F, v V = F(v) 8 e.g., [Zeidler 1995a, Th. 2.E] 20

24 2 variational theory of pdes for all v V. We recall that w ϕ w is a continuous linear mapping from V to V with operator norm 1. Thus, 0 = a(u, v) F(v) = ϕ u ϕ F, v V = 0 for all v V, which holds if and only if ϕ u = ϕ F in V. We now wish to solve this equation using the Banach fixed point theorem. 9 For δ > 0, consider the mapping T : V V, T(v) = v δ(ϕ v ϕ F ). If T is a contraction, then there exists a unique fixed point u such that T(u) = u and hence ϕ u ϕ F = 0. It remains to show that there exists a δ > 0 such that T is a contraction, i.e., there exists 0 < L < 1 with Tv 1 Tv 2 V L v 1 v 2 V. Let v 1, v 2 V be arbitrary and set v = v 1 v 2. Then we have Tv 1 Tv 2 2 V = v 1 v 2 δ(ϕ v1 ϕ v2 ) 2 V = v ϕ v 2 V = v 2 V 2δ v, ϕ v V + δ 2 ϕ v 2 V = v 2 V 2δa(v, v) + δ2 a(v, ϕ v ) v 2 V 2δc 1 v 2 V + δ2 c 2 v V ϕ v V (1 2δc 1 + δ 2 c 2 ) v 1 v 2 2 V. We can thus choose δ > 0 such that L 2 := (1 2δc 1 + δ 2 c 2 ) < 1, and the Banach fixed point theorem yields existence and uniqueness of the solution u V. To show the estimate (2.6), assume u 0 (otherwise the inequality holds trivially). Note that F is a bounded linear functional by assumption (ii), hence F V. We can then apply the coercivity of a and divide by u V 0 to obtain c 1 u V a(u, u) u V a(u, v) F(v) sup = sup = F v V v V v V v V. V We can now give sufficient conditions on the coefficients a jk, b j, c and d such that the boundary value problems defined above have a unique solution. Theorem 2.7 (Well-posedness). Let a jk L (Ω) satisfying the ellipticity condition (2.3) with constant α > 0, b j, c L (Ω) and f L 2 (Ω) and g L 2 ( Ω) be given. Set β = α 1 n j=1 b j 2. L (Ω) 9 e.g., [Zeidler 1995a, Th. 1.A] 21

25 2 variational theory of pdes a) The homogeneous Dirichlet problem has a unique solution u H 1 0 (Ω) if c(x) β 2 0 for almost all x Ω. In this case, there exists a C > 0 such that u H 1 (Ω) C f L 2 (Ω). Consequently, the inhomogeneous Dirichlet problem for g H 1 ( Ω) has a unique solution satisfying u H 1 (Ω) C( f L 2 (Ω) + g H 1 ( Ω) ). b) The Neumann problem for g L 2 ( Ω) has a unique solution u H 1 (Ω) if c(x) β 2 > 0 for almost all x Ω. In this case, there exists a C > 0 such that u H 1 (Ω) C( f L 2 (Ω) + g L 2 ( Ω) ). c) The Robin problem for g L 2 ( Ω) and d L ( Ω) has a unique solution if c(x) β 2 0 for almost all x Ω.d(x) 0 for almost all x Ω, and at least one inequality is strict. In this case, there exists a C > 0 such that u H 1 (Ω) C( f L 2 (Ω) + g L 2 ( Ω) ). Proof. We wish to apply the Lax Milgram theorem. Continuity of a and F follow by the Hölder inequality and the boundedness of the coefficients. It thus remains to verify the coercivity of a, which we only do for the case of homogeneous Dirichlet conditions (the others being similar). Let v H 1 0 (Ω) be given. First, the ellipticity of a jk implies that n Ω j,k=1 a jk j v(x) k v(x) dx α Ω n j v(x) 2 dx = α j=1 We then have by repeated application of Hölder s inequality j=1 n j v 2 L 2 (Ω) = α v 2 H 1 (Ω). ( n ) 1 a(v, v) α v 2 H 1 (Ω) 2 b j 2 L (Ω) v H1(Ω) v + L2(Ω) c(x)v(x) 2 dx Ω α 2 v 2 H 1 (Ω) + Ω ( c(x) 1 2α j=1 ) n b j 2 L (Ω) v 2 dx, j=1 22

26 2 variational theory of pdes where we have used Young s inequality ab α 2 a α b2 for a, b R and any α > 0. Under the assumption that c β 0, the second term is non-negative and we deduce using 2 Poincaré s inequality that a(v, v) α 2 v 2 H 1 (Ω) α 4 v 2 H 1 (Ω) + holds for C := α/(4 + 4c 2 Ω ). α 4c 2 Ω v 2 L 2 (Ω) C v 2 H 1 (Ω) Note that these conditions are not sharp; different ways of estimating the first-order terms in a give different conditions. For example,if b j W 1, (Ω),we can take β = n j=1 jb j L (Ω). Naturally, if the data has higher regularity, we can expect more regularity of the solution as well. The corresponding theory is quite involved, and we give only two useful results. Theorem 2.8 (Higher regularity 10 ). Let Ω R n be bounded domain with C k+1 boundary, k 0, a jk C k (Ω) and b j, c W k, (Ω). Then for any f H k (Ω), the solution of the homogeneous Dirichlet problem is in H k+2 (Ω) H 1 0 (Ω), and there exists a C > 0 such that u H k+2 (Ω) C( f H k (Ω) + u H 1 (Ω) ). Theorem 2.9 (Higher regularity 11 ). Let Ω be a convex polygon in R 2 or a parallelepiped in R 3, a jk C 1 (Ω) and b j, c C 0 (Ω). Then the solution of the homogeneous Dirichlet problem is in H 2 (Ω), and there exists a C > 0 such that u H 2 (Ω) C f L 2 (Ω). For non-convex polygons, u H 2 (Ω) is not possible. This is due to the presence of so-called corner singularities at reentrant corners, which severely limits the accuracy of finite element approximations. This requires special treatment, and is a topic of extensive current research. 10 [Troianiello 1987, Th. 2.24] 11 [Grisvard 1985, Th ], [Ladyzhenskaya and Ural tseva 1968, pp ] 23

27 Part II CONFORMING FINITE ELEMENT APPROXIMATION OF ELLIPTIC PDES

28 GALERKIN APPROACH FOR VARIATIONAL PROBLEMS We have seen that elliptic partial differential equations can be cast into the following form: Given a Hilbert space V, a bilinear form a : V V R and a continuous linear functional F : V R, find u V satisfying 3 (W) a(u, v) = F(v) for all v V. According to the Lax Milgram theorem, this problem has a unique solution if there exist c 1, c 2 > 0 such that (3.1) a(v, v) c 1 v 2 V, (3.2) a(u, v) c 2 u V v V hold for all u, v V (which we will assume from here on). The conforming Galerkin approach consists in choosing a (finite-dimensional) closed subspace V h V and looking for u h V h satisfying 1 (W h ) a(u h, v h ) = F(v h ) for all v h V h. Since we have chosen a closed V h V, the subspace V h is a Hilbert space with inner product, V. Furthermore, the conditions (3.1) and (3.2) are satisfied for all u h, v h V h as well. The Lax Milgram theorem thus immediately yields the well-posedness of (W h ). Theorem 3.1. Under the assumptions of Theorem 2.6, for any closed subspace V h V, there exists a unique solution u h V h of (W h ) satisfying u h V 1 c 1 F V. 1 The subscript h stands for a discretization parameter, and indicates that we expect convergence of u h to the solution of (W) as h 0. 25

29 3 galerkin approach for variational problems The following result is essential for all error estimates of Galerkin approximations. Lemma 3.2 (Céa s lemma). Let u h be the solution of (W h ) for given V h V and u be the solution of (W). Then, u u h V c 2 inf u v h c V, 1 v h V h where c 1 and c 2 are the constants from (3.1) and (3.2). Proof. Since V h V, we deduce (by subtracting (W) and (W h ) with the same v V h ) the Galerkin orthogonality a(u u h, v h ) = 0 for all v h V h. Hence, for arbitrary v h V h, we have v h u h V h and therefore a(u u h, v h u h ) = 0. Using (3.1) and (3.2), we obtain c 1 u u h 2 V a(u u h, u u h ) = a(u u h, u v h ) + a(u u h, v h u h ) c 2 u u h V u v h V. Dividing by u u h V, rearranging, and taking the infimum over all v h V h yields the desired estimate. This implies that the error of any (conforming) Galerkin approach is determined by the approximation error of the exact solution in V h. The derivation of such error estimates will be the topic of the next chapters. the symmetric case The estimate in Céa s lemma is weaker than the corresponding estimate (1.1) for the model problem in Chapter 1. This is due to the symmetry of the bilinear form in the latter case, which allows characterizing solutions of (W) as minimizers of a functional. Theorem 3.3. If a is symmetric, u V satisfies (W) if and only if u is the minimizer of over all v V. Proof. For any u, v V and t R, J(v) := 1 a(v, v) F(v) 2 J(u + tv) = J(u) + t(a(u, v) F(v)) + t2 a(v, v) 2 26

30 3 galerkin approach for variational problems due to the symmetry of a. Assume that u satisfies a(u, v) F(v) = 0 for all v V. Then, setting t = 1, we deduce that for all v 0, J(u + v) = J(u) + 1 a(v, v) > J(u) 2 holds. Hence, u is the unique minimizer of J. Conversely, if u is the (unique) minimizer of J, every directional derivative of J at u must vanish, which implies for all v V. 0 = d dt J(u + tv) t=0 = a(u, v) F(v) Together with coercivity and continuity, the symmetry of a implies that a(u, v) is an inner product on V that induces an energy norm u a := a(u, u) 1 2. (In fact, in many applications, the functional J represents an energy which is minimized in a physical system. For example in continuum mechanics, 1 2 u 2 a = 1 a(u, u) represents the elastic deformation energy of a 2 body, and F(v) its potential energy under external load.) Arguing as in Chapter 1.2, we see that the solution u h V h of (W h ) which is called Ritz Galerkin approximation in this context satisfies u u h a = min v h V h u v h a, i.e., u h is the best approximation of u in V h in the energy norm. Equivalently, one can say that the error u u h is orthogonal to V h in the inner product defined by a. Often it is more useful to estimate the error in a weaker norm. This requires a duality argument. Let H be a Hilbert space with inner product (, ) and V a closed subspace satisfying the conditions of the Lax Milgram theorem theorem such that the embedding V H is continuous (e.g., V = H 1 L 2 = H). Then we have the following estimate. Lemma 3.4 (Aubin Nitsche lemma). Let u h be the solution of (W h ) for given V h V and u be the solution of (W). Then, there exists a C > 0 such that ( ) 1 u u h H C u u h V sup inf ϕ g v h g H g v H h V V h holds, where for given g H, ϕ g is the unique solution of the adjoint problem a(w, ϕ g ) = (g, w) for all w V. Since a is symmetric, the existence of a unique solution of the adjoint problem is guaranteed by the Lax Milgram theorem. 27

31 3 galerkin approach for variational problems Proof. We make use of the dual representation of the norm in any Hilbert space, (g, w) (3.3) w H = sup, g H g H where the supremum is taken over all g 0. Now, inserting w = u u h in the adjoint problem, we obtain for any v h V h using the Galerkin orthogonality and continuity of a that (g, u u h ) = a(u u h, ϕ g ) Inserting w = u u h into (3.3), we thus obtain = a(u u h, ϕ g v h ) C u u h V ϕ g v h V. (g, u u h ) u u h H = sup g H g H C u u h V sup g H ϕ g v h V g H for arbitrary v h V h, and taking the infimum over all v h yields the desired estimate. The Aubin Nitsche lemma also holds for nonsymmetric a, provided both the original and the adjoint problem satisfy the conditions of the Lax Milgram theorem (e.g., for constant coefficients b j ). 28

32 FINITE ELEMENT SPACES Finite element methods are a special case of Galerkin methods, where the finite-dimensional subspace consists of piecewise polynomials. To construct these subspaces, we proceed in two steps: 1. We define a reference element and study polynomial interpolation on this element. 2. We use suitably modified copies of the reference element to partition the given domain and discuss how to construct a global interpolant using local interpolants on each element. We then follow the same steps in proving interpolation error estimates for functions in Sobolev spaces construction of finite element spaces To allow a unified study of the zoo of finite elements proposed in the literature, 1 we define a finite element in an abstract way. Definition 4.1. Let (i) K R n be a simply connected bounded open set with piecewise smooth boundary (the element domain, or simply element if there is no possibility of confusion), (ii) P be a finite-dimensional space of functions defined on K (the space of shape functions), (iii) N = {N 1,..., N d } be a basis of P (the set of nodal variables or degrees of freedom). Then (K, P, N) is a finite element. 1 For a far from complete list of elements, see, e.g., [Brenner and Scott 2008, Chapter 3], [Ciarlet 2002, Section 2.2] 29

33 4 finite element spaces As we will see, condition (iii) guarantees that the interpolation problem on K using functions in P and hence the Galerkin approximation is well-posed. The nodal variables will play the role of interpolation conditions. This is a somewhat backwards definition compared to our introduction in Chapter 1 (where we have directly specified a basis for the shape functions). However, it leads to an equivalent characterization that allows much greater freedom in defining finite elements. The connection is given in the next definition. Definition 4.2. Let (K, P, N) be a finite element. The basis {ψ 1,..., ψ d } of P dual to N, i.e., satisfying N i (ψ j ) = δ ij, is called nodal basis of P. For example, for the linear finite elements in one dimension, K = (0, 1), P is the space of linear polynomials, and N = {N 1, N 2 } are the point evaluations N 1 (v) = v(0), N 2 (v) = v(1) for every v P. The nodal basis is given by ψ 1 (x) = 1 x and ψ 2 (x) = x. Condition (iii) is the only one that is difficult to verify. The following Lemma simplifies this task. Lemma 4.3. Let P be a d-dimensional vector space and let {N 1,..., N d } be a subset of P. Then, the following statements are equivalent: a) {N 1,..., N d } is a basis of P, b) If v P satisfies N i (v) = 0 for all 1 i d, then v = 0. Proof. Let {ψ 1,..., ψ d } be a basis of P. Then, {N 1,..., N d } is a basis of P if and only if for any L P, there exist (unique) α i, 1 i d such that L = d α j N j. j=1 Using the basis of P, this is equivalent to L(ψ i ) = d j=1 α jn j (ψ i ) for all 1 i d. Let us define the matrix B = (N j (ψ i )) d i,j=1 and the vectors L = (L(ψ 1 ),..., L(ψ d )) T, a = (α 1,..., α d ) T. Then, (a) is equivalent to Ba = L being uniquely solvable, i.e., B being invertible. On the other hand, given any v P, we can write v = d j=1 β jψ j. The condition (b) can be expressed as n β j N i (ψ j ) = N i (v) = 0 j=1 for all 1 i d implies v = 0, or, in matrix form, that B T b = 0 implies 0 = b := (β 1,..., β d ) T. But this too is equivalent to the fact that B is invertible. 30

34 4 finite element spaces Note that (b) in particular implies that the corresponding interpolation problem is uniquely solvable. Another useful tool is the following Lemma, which is a multidimensional form of polynomial division. Lemma 4.4. Let L 0 be a linear function on R n and P be a polynomial of degree d 1 with P(x) = 0 for all x with L(x) = 0. Then, there exists a polynomial Q of degree d 1 such that P = LQ. Proof. First, we note that affine transformations map the space of polynomials of degree d to itself. Thus, we can assume without loss of generality that P vanishes on the hyperplane orthogonal to the x n axis, i.e. L(x) = x n and P(^x, 0) = 0, where ^x = (x 1,..., x n 1 ). Since the degree of P is d, we can write P(^x, x n ) = d c α,j^x α x j n j=0 α d j for a multi-index α N n 1 and ^x α = x α 1 1 xα n 1 n 1. For x n = 0, this implies 0 = P(^x, 0) = and therefore c α0 = 0 for all α d. Hence, where Q is of degree d 1. P(^x, x n ) = d α d j=1 α d j = x n d j=1 α d j =: x n Q = LQ, c α,0^x α, c α,j^x α x j n c α,j^x α x j 1 n 4.2 examples of finite elements We restrict ourselves to the case n = 2 (higher dimensions being similar) and the most common examples. 31

35 4 finite element spaces z 3 z 3 z 3 L 2 L 1 z 5 z 4 z 4 z 1 z 2 L 3 (a) Linear Lagrange element z 1 z 6 z 2 (b) Quadratic Lagrange element z 1 z 2 (c) Cubic Hermite element Figure 4.1: Triangular finite elements. Filled circles denote point evaluation, open circles gradient evaluations. triangular elements Let K be a triangle and { } P k = α k c αx α : c α R denote the space of all bivariate polynomials of total degree less than or equal k, e.g., P 2 = span {1, x 1, x 2, x 2 1, x2 2, x 1x 2 }. It is straightforward to verify that P k (and hence Pk) is a vector space of dimension 1 (k + 1)(k + 2). We consider two types of interpolation conditions: function values (Lagrange interpolation) and gradient values (Hermite interpolation). The following 2 examples define valid finite elements. Note that the argumentation is essentially the same as for the well-posedness of the corresponding one-dimensional polynomial interpolation problems. Linear Lagrange elements. Let k = 1 and take P = P 1 (hence the dimension of P and P is 3) and N = {N 1, N 2, N 3 } with N i (v) = v(z i ), where z 1, z 2, z 3 are the vertices of K (see Figure 4.1a). We need to show that condition (iii) holds, which we will do by way of Lemma 4.3. Suppose that v P 1 satisfies v(z 1 ) = v(z 2 ) = v(z 3 ) = 0. Since v is linear, it must also vanish on each line connecting the vertices, which can be defined as the zero-sets of the linear functions L 1, L 2, L 3. Hence, by Lemma 4.4, there exists a constant (i.e., polynomial of degree 0) c such that v = cl 1. Now let z 1 be the vertex not on the edge defined by L 1. Then, 0 = v(z 1 ) = cl 1 (z 1 ), and therefore c = 0 and so v = 0 (since L 1 (z 1 ) 0). Quadratic Lagrange elements: Let k = 2 and take P = P 2 (hence the dimension of P and P is 6). Set N = {N 1, N 2, N 3, N 4, N 5, N 6 } with N i (v) = v(z i ), where z 1, z 2, z 3 are again the vertices of K and z 4, z 5, z 6 are the midpoints of the edges described by the linear functions L 1, L 2, L 3, respectively (see Figure 4.1b). To show that condition (iii) holds, we argue as above. Let v P 2 vanish at z i, 1 i 6. On each edge, v 32