Variational Formulations

Transcription

1 Chapter 2 Variational Formulations In this chapter we will derive a variational (or weak) formulation of the elliptic boundary value problem (1.4). We will discuss all fundamental theoretical results that provide a rigorous understanding of how to solve (1.4) using the finite element method. 2.1 Computational domains We shall consider so-called Lipschitz domains as computational domains. An open, bounded, connected set R d, d = 1, 2, 3, is called Lipschitz domain, if its boundary := \ is a Lipschitz boundary, which is roughly speaking the case if there exists a finite covering of with open subsets of R d (rectangles), and is locally (in each open rectangle) the graph of a Lipschitz continuous function. This particularly implies that the boundary of a Lipschitz domain is of finite length (excluding fractal boundaries), and slightly smoother than continuous, i. e. excluding slit and cusp domains. Figure 2.1: Domains that are not Lipschitz: slit domain (left), cusp domain (middle) and domain whose boundary is the limit of a Koch curve and has infinite length (right). In almost all numerical computations only a special type of Lipschitz domains are relevant, namely domains that can be described in the widely used CAD (computer-aided design) software packages. This set of domains is usually identical to the domains that we shall define in the following. Definition 2.1. In the case d = 2 a connected domain is called a curvilinear Lipschitz polygon, if is a Lipschitz domain, and there is a finite number of open subsets Γ k, k = 1,..., P, P N (where open refers to the d 1-dimensional manifold of R d ) such that := Γ 1 Γ P, Γ k Γ l = if k l, and for each k {1,..., P } there is a C 1 -diffeomorphism ( invertible mapping of smooth manifolds) Φ k : [0, 1] Γ k. The boundary segments are called edges, their endpoints are the vertices of. 11

2 A tangential direction can be defined for all points of an edge, which allows us to define angles of the vertices, see Figure 2.2. Analogously, a normal direction can be defined, which implies that the exterior unit vectorfield n : R d is well-defined almost everywhere on. The mappings Φ k can be regarded as smooth parametrizations of the edges. Figure 2.2: Curvilinear Lipschitz polygon with added angles at vertices. An analogous notion exists in three dimensions: curved Lipschitz polyhedron. Definition 2.2. A subset R d is called a computational domain if it is bounded and its boundary is of class C k, k 1, or if it is a bounded connected interval for d = 1, a curvilinear Lipschitz polygon for d = 2, a curved Lipschitz polyhedron for d = Linear differential operators Let α N d 0 be a multi-index, i. e., a vector of d non-negative integers α = (α 1,..., α d ) N d 0. We set α := α α d and call α α := α1 d x α1 1 x α d n the partial derivative of order α. Remember that for sufficiently smooth functions all partial derivatives commute. Provided that the derivatives exist, the gradient of a function f : R d R is the column vector ( f f(x) := (x),..., f ) (x), x. x 1 x d The divergence of a vector field f = (f 1,..., f d ) : R d R d is the function f(x) := d k=1 f k x k (x), x. The differential operator := is known as Laplacian. 2.3 Integration by parts Below we assume that R d, d = 1, 2, 3, is bounded and an interval for d = 1, a Lipschitz polygon for d = 2, and a Lipschitz polyhedron for d = 3. Throughout this section we adopt the notation n = (n 1,..., n d ) T for the exterior unit normal vectorfield that is defined almost everywhere on Γ :=, see Section

3 Theorem 2.3 (Gauß theorem). If f (C 1 ()) d (C 0 ()) d, then f dx = f, n ds. By the product rule (u f) = u f + u, f for u C 1 (), f (C 1 ()) d, we deduce the first Green s formula f, u + f u dx = f, n uds Γ Γ (FGF) for all f (C 1 ()) d (C 0 ()) d and all u C 1 () C 0 (). Plugging in the special f = fe k, k = 1,..., d, where e k is the k-th unit vector, we get f u + f u dx = fu (n) k ds (2.1) x k x k for f, u C 1 () C 0 (). We may also plug f = v into (FGF), which yields v, u + v u dx = v, n u ds (2.2) for all v C 2 () C 1 (), u C 1 () C 0 (). 2.4 Distributional derivatives Definition 2.4 (Test function space). For a non-empty open set R d we denote C 0 () := {v C () : supp v := {x : v(x) 0} } the space of test functions, which is the space of functions C () with compact support. Lemma 2.5. Two integrable functions f and g defined in the bounded set are almost everywhere equal if and only if fv dx = gv dx for all test functions v C 0 (). We consider in the following functions to be equivalent if they are equal almost everywhere. Definition 2.6. Let u L 2 () and α N d 0. A function w L 2 () is called the weak derivative or distributional derivative α u (of order α ) of u, if wv dx = ( 1) α u α v dx v C0 (). Based on this definition, all linear differential operators introduced in Section 2.2 can be given a weak/distributional interpretation. For example, the weak gradient u of a function u L 2 () will be a vectorfield w (L 2 ()) d with w, v dx = u v dx v (C0 ()) d. (2.3) This can be directly concluded from (FGF). The same is true of the weak divergence w L 2 () of a vectorfield u (L 2 ()) d w v dx = u, v dx v C0 (). (2.4) Theorem 2.7. If u C m (), then all weak derivatives of order m agree in L 2 () with the corresponding classical derivatives. Proof. Clear by a straightforward application of (2.1). Hence, without changing notations, all derivatives will be understood as weak derivatives in the sequel. Straight from the definition we also infer that all linear differential operators in weak sense commute. Γ Γ 13

4 Extension of integration by parts formula For Lipschitz domains the Gauß theorem (Theorem 2.3) and hence, also the first Green s formula (FGF) can be extended to integrable functions with weak derivatives. Weak derivative in subdomains The weak derivative in and a subdomain 1 coincide in 1 since the corresponding spaces of test functions satisfy C 0 ( 1 ) C 0 (). Continuity over interfaces of subdomains For the partition = 1 2, 1 2 =, where both subdomains are supposed to have a Lipschitz boundary, a function u, which possesses a weak gradient in 1 and 2, possesses a weak gradient in if and only if [u] Σ = 0 on the common interface Σ := 1 2. Proof. The only candidate for weak gradient in can only be the piecewise weak gradient in 1 and 2. Let n Σ denote the normal vector on Σ pointing outwards 1. We test with v (C0 ()) d and so u v dx = u v dx 1 2 = u v dx + (u 1 v n Σ u 2 v n Σ )ds(x) 1 2 Σ = u v dx, if and only if Σ [u] Σ vds(x) = 0 v C 0 (Σ), which is by Lemma 2.5 equivalent to [u] Σ = 0 almost everywhere. Remark 2.8. The fact that a function has classical derivatives in subdomains and even the fact that the classical derivatives can be extended to a function in, which is in L 2 (), does not guarantee that it has a weak derivative in the whole domain. For example a function which is piecewise C ( l ), l = 1, 2, but discontinuous over Σ := 1 2, does not have a weak gradient. 2.5 Variational formulations of elliptic boundary value problems Dirichlet boundary conditions Interpreting the divergence in (1.4a) as weak derivative we can equivalently write a u, v + c uv dx = fv dx (2.5) for test functions v C 0 (). Additionally the solution u has to fulfill the Dirichlet boundary conditions (1.4b), which all trial functions functions to try if they solve (2.5) have to fulfill. Boundary conditions which are present in constraint to the space of trial or test functions are called essential Neumann boundary conditions We enrich the space of test functions and choose v C () C(). Using (FGF) we get a u, v + c uv dx a u, n v ds = fv dx (2.6) for all v C () C(). We can incorporate the Neumann boundary condition (1.4c) in the variational formulation. For pure Neumann boundary conditions, i. e., Γ N =, this reads a u, v + c uv dx = fv dx + h v ds (2.7) 14

5 for all v C () C(). Boundary conditions which are present in the variational formulation are called natural. To see that (2.7) is equivalent to (1.4a) and (1.4c) we choose first test functions v C0 (). Then, the term on disappears, and with the definition of the weak divergence (2.4) we get (1.4a). Now, let v C () C 0 ( ). Integrating by parts and using the fact that (1.4a) holds we get a u, n v ds = hvds which is by Lemma 2.5 equivalent to (1.4c) Robin boundary conditions For the case of pure Robin boundary conditions, where Neumann boundary conditions are included with β = 0, inserting (1.4d) into (2.6) leads to a u, v + c uv dx + βu v ds = fv dx + h v ds. (2.8) Also Robin boundary conditions are natural. 2.6 Linear and bilinear forms Let V, W be real/complex vector spaces. Linear operator is a mapping of the form T : V W, that satisfies T(λv + µw) = λ T v + µ T w for all v, w V, λ, µ R/C. Linear form is a linear operator of the form l : V R/C. Bilinear form is a mapping of the form b : V V R/C, where v b(v, w) for every w and w b(v, w) for every v are both linear forms on V, e. g. a u, v + c uv dx = fv dx } {{ } }{{ } b(u,v) l(v) Antilinear operator is a mapping of the form T : V W that satisfies T(λv + µw) = λ T v + µ T w for all v, w V, λ, µ R/C. Antilinear form is an antilinear operator of the from l : V R/C. Sesquilinear form is a mapping of the form b : V V R/C, where v b(v, w) is a linear form for every w and w b(v, w) is an antilinear form for every v, e. g. a u, v + c uv dx = fv dx } {{ } }{{ } b(u,v) l(v) Linear variational problem Given some linear form l seek u V such that b(u, v) = l(v) v V. (LVP) Symmetric bi-/sesquilinear form if b(v, w) = b(w, v) for all v, w V. Positive definite bi-/sesquilinear form if for all v V we have b(v, v) > 0 v 0. 15

6 V -continuous bi-/sesquilinear form satisfies b(u, v) b u V v V for all u, v V with continuity constant b := sup u,v V \{0} u V v V b(u,v). (This property is also called boundedness, i. e., b is said to be bounded in V with constant b.) V -elliptic bi-/sesquilinear form satisfies b(v, v) γ v 2 V for all v V with ellipticity constant γ. (This property is also called coercivity.) A symmetric positive definite bi-/sesquilinear form b is an inner product that induces a norm through The fundamental Cauchy-Schwarz inequality v b := b(v, v) 1 2 v V. b(v, w) v b w b v, w V ensures that b is continuous with continuity constant 1 with respect to its induced norm. Moreover, we have Pythagoras theorem a(v, w) = 0 v 2 a + w 2 a = v + w 2 a. In the context of elliptic partial differential equations a norm that can be derived from a V -elliptic bi-/sesquilinear form is often dubbed energy norm, denoted by e. 2.7 Sobolev spaces Definition 2.9. A normed vector space V is complete, if every Cauchy sequence {v k } k V has a limit v in V. A complete normed vector space is called a Banach space. Definition A Hilbert space is a Banach space whose norm is induced by an inner product. In Section 2.5 we learned that the formal variational problem associated with the pure homogeneous Neumann problem for (1.4) is: seek u : R such that a u, v + c uv dx = fv dx (2.9) for all test functions v. Ideal space of u and v in (2.9) is a Hilbert space H whose inner product coincides with the bi-/sesquilinear form, and consequently, which is equipped with the energy norm v 2 e := a v 2 + c v 2 dx and the space is defined as H := {v : R : weak gradient v exists and v e < }. The existence of a weak gradient u is built in the space, so we restrict the search only to those functions. The same space applies for test functions, that are now more functions than C () (equivalence due to density by Meyers-Serrin theorem, see Theorem 2.15). Definition 2.11 (Sobolev space H 1 ()). The Sobolev space H 1 () is the space of square integrable functions R with square integrable weak gradients with norm H 1 () := {v L 2 () : (the weak gradient) v exists and v L 2 ()} v 2 H 1 () := v 2 L 2 () + D2 v 2 L 2 () where D = diam() is introduced to match units, which is often omitted in the definition. 16

7 Note, that the H 1 ()-norm and the energy norm are equivalent if the latter is based on an H 1 ()- elliptic and H 1 ()-continuous bilinear form, i. e., there exist two constants C 1, C 2 > 0 such that C 1 v e v H 1 () C 2 v e v H 1 (). Definition For m N 0 and R d we define the Sobolev space of order m as H m () := {v L 2 () : α v L 2 (), α m}, equipped with the norm v Hm () := ( α m α v 2 L 2 () ) 1 2. Notation For all m N 0 and R d denotes a semi-norm on H m (). v H m () := ( α =m α v 2 L 2 () ) 1 2 Theorem The Sobolev spaces H m (), m N 0, are Hilbert spaces with the inner product (u, v) Hm () := ( α u, α v) L2 () u, v H m (). α m Theorem 2.15 (Meyers-Serrin theorem). The space C () is a dense subspace of H m () for all m N 0. Definition H 1 0 () is defined as the completion of C 0 () with respect to the H 1 ()-norm. 2.8 Theory of variational formulations Ellipticity and the lemma of Lax-Milgram Definition 2.17 (Ellipticity of bilinear forms). A bilinear form b(, ) is called V -elliptic, if there exists a γ > 0 such that for all v V b(v, v) γ v 2 V. Definition 2.18 (Ellipticity of sesquilinear forms). A sesquilinear form b(, ) is called V -elliptic, if there exists a γ > 0 and a θ C with θ = 1, such that for all v V Re(θ b(v, v) γ v 2 V. we rotate the sesquilinear form by θ to have positive real part Definition The dual V of a normed vector space V is the normed vector space of continuous linear forms on V. The dual space is equipped with the (operator) norm f V = sup 0 v V f(v) v V. Example Linear forms fvdx with f L2 () are in the dual of H 1 () since by the Cauchy- Schwarz inequality sup fvdx f L sup 2 () v L 2 () f L 0 v H 1 () v H1 () 0 v H 1 () v 2 (). H1 () Lemma 2.21 (Lax-Milgram). Let V be a reflexive Banach space. Let the bilinear form b : V V R or sesquilinear form b : V V C be bounded and V -elliptic. Then, the variational problem (LVP) has for any f V a unique solution u V with u V 1 γ f V. 17

8 Proof. The V -ellipticity of the sesquilinear form b implies for u V γ u 2 V Re(θb(u, u)) θb(u, u) = b(u, u) = f(u) f V u V, and so the stability estimate and the uniqueness (f = 0 u = 0) follow. The existence follows by the Riesz representation theorem where the bi-/sesquilinear form is taken as inner product. Theorem 2.22 (Riesz representation theorem). Let H be a Hilbert space and ϕ H an arbitrary linear form on H. Then there exists a unique element u H such that Moreover, ϕ H = u H The inf-sup conditions ϕ(v) = (u, v) H for all v H. A more general result on the existence and uniqueness of weak solutions is provided by the inf-sup conditions. Theorem Let U, V be a reflexive Banach spaces with norms U and V. Furthermore let the bi-/sesquilinear form b be bounded in U, V. Then the following statements are equivalent: (i) For all f V the linear variational problem (LVP) has a unique solution u f U that satisfies with γ > 0 independent of f. u f U 1 γ f V, (ii) The bilinear form b satisfies the inf-sup conditions γ > 0 : inf sup w U\{0} v V \{0} b(w, v) v V w U γ, (IS1) v V \ {0} : sup b(w, v) > 0. (IS2) w U\{0} 2.9 Well-posedness of elliptic boundary value problems Boundary value problems with Dirichlet data For the variational formulation of the PDE (1.4) the Dirichlet boundary conditions are essential and are considered in the space of trial functions H 1 Γ D,g() := {v H 1 () : v = g on Γ D } H 1 (), which is equipped with the (usual) H 1 ()-norm. Note that HΓ 1 D,g() is not a vector space, i. e., if g 0, u HΓ 1 D,g () and λ 1 then λu H1 Γ D,g (). For the test functions we use the associated homogeneous space H 1 Γ D,0 (). If Γ D = we can write shortly H 1 g () and H 1 0 (). Variational formulation for pure Dirichlet boundary conditions: Seek u Hg 1 () such that a(x) u(x) v(x) + cu(x)v(x)dx = f(x)v(x)dx v H0 1 (). It is equivalent to write u = u 0 + u g with u g Hg 1 () some extension of g into and the new unknown (offset) u 0 H0 1 (): Seek u 0 H0 1 () such that b(u 0, v) = a u 0 v + cu 0 vdx = fv a u g v + cu g vdx = l(v) v H0 1 (). For well-posedness it is enough to prove H 1 0 ()-ellipticity b(v, v) γ v H1 () v H 1 0 (), which is a weaker statement than H 1 ()-ellipticity. 18

9 Case c c 0 > 0 Since by assumption a a 0 > 0, we have b(v, v) min{a 0, c 0 } v H 1 () v H 1 (), from which H 1 0 ()-ellipticity directly follows. Case c = 0 Now we only have b(v, v) a 0 v L2 () v H 1 (). However, we shall apply the following result to obtain H 1 Γ D,0 ()-ellipticity. Lemma 2.24 (Friedrichs inequality). Let be an open bounded domain and Γ D > 0. Then, for all u H 1 Γ D,0 () there exists a constant C F (, Γ D ) such that Now we can conclude for all v H 1 Γ D,0 (). u 2 L 2 () C F (, Γ D ) diam() 2 u 2 L 2 (). b(v, v) a 0 v 2 L 2 () a 0 2 v 2 L 2 () + a 0 2C F diam() 2 v 2 L 2 () a { } 0 2 min 1 1, C F diam() 2 v 2 H 1 () For which functions g can we find an extension u g H 1 ()? Any restriction trace of a function v H 1 () to Γ D can be extended to v H 1 (). We can choose u g to be any of the functions v for which v = g on Γ D. For Γ D = the trace space of H 1 () is H 1 /2 ( ) L 2 ( ) with the norm v H 1 /2 ( ) = inf{ w H 1 () : w C (), w Γ = v}. The index 1 /2 corresponds to the definition of Sobolev spaces with rational indices (as interpolation spaces), where a direct definition of the norm exists (Sobolev-Slobodeckij norm). For all functions in g H 1 /2 ( ) an extension u g H 1 () exists, but not for any g L 2 ( ) (e. g., discontinuous functions). Theorem The trace operator R 1 : H 1 () H 1 /2 (Γ) is continuous and surjective (it is onto) and has a bounded right inverse (extension operator) E 1 : H 1 /2 (Γ) H 1 (), i. e., R 1 E 1 = Id Boundary value problems with Neumann data For the variational formulation of the boundary value problem (1.4) the Neumann boundary conditions are natural and we use in case of pure Neumann boundary conditions H 1 () as space of trial and test functions: Seek u H 1 () such that b(u, v) = a(x) u(x) v(x) + c(x)u(x)v(x)dx = f(x)v(x)dx + h(x)v(x)ds(x) = l(v) v H 1 (). (2.10) Analogously to the discussion of boundary value problems with Dirichlet data in Section we have H 1 ()-ellipticity if a a 0 > 0 and c c 0 > 0. 19

10 Case c 0 Let us test (2.10) with v = 1. Then b(u, 1) = 0 for all u H 1 (), and hence, l(1) = 0 must hold as well. This leads us to the compatibility condition for the data of the problem f dx + h ds = 0. (2.11) The solution is not unique. Since b(1, v) = 0 for any v H 1 () we can add any constant. The constant can be fixed by demanding a vanishing mean value of u, i. e., we switch to the subspace H () 1 := {v H 1 () : v dx = 0}. The variational formulation is: Seek u H 1 () such that b(u, v) = l(v) v H 1 (). The bilinear form b is H 1 ()-elliptic. For the proof we apply Lemma 2.26 (Poincaré s inequality). Let be a open bounded domain. Then, for all u H 1 () there exists a constant C P () such that where u = 1 u(x)dx is the mean of u in. u u 2 L 2 () C P () diam() 2 u 2 L 2 (), Case c c 0 > 0 in some subdomain c There is no need of a compatibility condition (2.11) since there exist functions u H 1 () such that b(u, 1) = c c(x)u(x)dx 0 The constant is also fixed by the term c c(x)u(x)v(x)dx as we can apply Lemma Let f : H 1 () R be a continuous quadratic functional which fixes constants and for which we have f(λv) = λ 2 f(v) for λ R, i. e., any constant function vanishes if f(1) = 0. Then, it exists a constant C = C(f, ) > 0 such that for all u H 1 () we have ( ) u 2 L 2 () C u 2 L 2 () + f(u). Since c c 0 > 0 in c, the (quadratic) functional c c(x) u(x) 2 dx fixes constants. Then, we have H 1 ()-ellipticity, since { b(v, v) a 0 v 2 L 2 () + c(x) v(x) 2 dx min 1, a } ( ) 0 2 v 2 L c 2 2 () + c(x) v(x) 2 dx c { min 1, a } ( 0 v 2 L 2 2 () + 1 C v 2 L ()) γ v 2 2 H 1 () for all v H 1 () where γ = min { 1, 1 C, a0 2, a0 2C } Boundary value problems with Robin data Variational formulation for pure Robin boundary conditions: Seek u H 1 () such that b(u, v) = a(x) u(x) v(x) + c(x)u(x)v(x)dx + = f(x)v(x)dx β(x)u(x)v(x)ds(x) h(x)v(x)ds(x) = l(v) v H 1 (). Analogously to above, we can deduce H 1 ()-ellipticity if a a 0 > 0 and c c 0 > 0 (like for Dirichlet boundary conditions) or if a a 0 > 0, c 0 and β β 0 > 0 (like for Neumann boundary conditions with c 0 in c ). 20

11 2.10 Discrete variational formulations A first step towards finding a practical algorithm for the approximate solution of (LVP) is to convert it into a discrete variational problem. The attribute discrete means that the solution can be characterized by a finite number of real (or complex) numbers. Galerkin discretization of (LVP) We replace V, e. g., H 1 () or H 1 0 (), in (LVP) by finite dimensional subspaces equipped with the same norm. The most general approach relies on two subspaces of V, i. e., If W n V : trial space, dim W n = N N, and V n V : test space, dim V n = N N. W n = V n we speak of a classical Galerkin discretization, if furthermore the bilinear form provides an inner product, we call it Ritz-Galerkin discretization, W n V n the discretization is called Petrov-Galerkin discretization. The subscript n labels discrete entities, but it can also deal as index referring to sequences of trial and test spaces in a convergence study. Discrete variational problem corresponding to (LVP): Seek u n W n such that b(u n, v n ) = l(v n ) v n V n. (DVP) Galerkin orthogonality of the discretization error e n := u u n to the test space V n We can insert v n V n into (DVP) and (LVP), the difference gives b(u u n, v n ) = 0 v n V n. The error is the smallest possible when measured in the energy norm (b-optimality). u e n V 0 u n V n Figure 2.3: b-orthogonality of the error e n = u u n with respect to V n, if b is an inner product. Well-posedness of the discretized problem For a classical Galerkin discretization with V n V V n -ellipticity follows directly from the V - ellipticity with the same constant γ and hence, by the Lax-Milgram lemma, existence and uniqueness follow. The error can be estimated by Lemma 2.28 (Céa s lemma). Let the bi-/sesquilinear form b be V -elliptic with constant γ and V - continuous with constant b. Then, the solutions u of (LVP) and u n of (DVP) satisfy the quasioptimality estimate u u n V b γ inf u v n V. v n V n 21

12 Proof. Using V -ellipticity, Galerkin orthogonality and continuity of b we have where v n V n was arbitrarily chosen. γ u u n 2 V b(u u n, u u n ) = b(u u n, u v n + v n u n ) b(u u n, u v n ) + b(u u n, v n u n ) }{{} =0 by Galerkin orthog. b u u n V u v n V For the energy norm we have γ = b = 1 and thus what is known as b-optimality. u u h b inf v n V n u v n b For general bi-/sesquilinear forms we need besides (IS1), (IS2) that the discrete inf-sup-condition is fulfilled γ n > 0 : inf sup w n W n v n V n\{0} b(w n, v n ) v n V w n V γ n. (DIS) One cannot conclude (DIS) from (IS1) and (IS2) because the supremum is taken over a much smaller set. If the constant γ n is uniformly bounded away from zero an asymptotic quasi-optimality of the Galerkin solution is guaranteed ( u u n V 1 + b ) V V R inf u w n γ n w V, n W n which is automatically satisfied in case of ellipticity, see Céa s lemma (Lemma 2.28) Algebraic setting (DVP) can be solved on a computer by introducing ordered bases B V := {p 1 n,..., p N n } of V n, B W := {q 1 n,..., q N n } of W n, N := dim V n = dim W n. Linear system of equations Bu = f (LSE) with B := ( b(q k n, p j n) ) N j,k=1 RN,N, f := ( l(p k n) ) N k=1 RN. Unknown are the coefficients u (coefficient vector) in the basis representation. Equivalence: (LSE) has a unique solution if and only if (DVP) has a unique solution. j is the row index, whereas k is the column index: element b(q k n, p j n) of the matrix B is highlighted: j k 22

13 (Continuous) variational problem (LVP): b(u, v) = l(v) Choose finite dimensional trial and test spaces W n and V n. Discrete variational problem (DVP): b(u n, v n ) = l(v) Choose bases of W n and V n Algebraic problem (LSE): Bu = f Figure 2.4: Overview of stages involved in the Galerkin discretization of an abstract variational problem. 23

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