Multiscale techniques for parabolic equations

We use the local orthogonal decomposition technique introduced in Målqvist and Peterseim (Math Comput 83(290):2583–2603, 2014) to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale coefficients. We consider nonsmooth initial data and a backward Euler scheme for the temporal discretization. Optimal order convergence rate, depending only on the contrast, but not on the variations of the coefficients, is proven in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\infty (L_2)$$\end{document}L∞(L2)-norm. We present numerical examples, which confirm our theoretical findings.


Introduction
In this paper we study numerical solutions to parabolic equations with highly varying coefficients. These equations appear, for instance, when modeling physical behavior in a composite material or a porous medium. Such problems are often referred to as multiscale problems.  Convergence of optimal order of classical finite element methods (FEMs) based on continuous piecewise polynomials relies on at least spatial H 2 -regularity. More precisely, for piecewise linear polynomials, the error bound depends on u H 2 , which may be proportional to −1 if the diffusion coefficient varies on a scale of . Thus, the mesh width h typically must fulfill h < to achieve convergence. However, this is not computationally feasible in many applications. To overcome this issue, several numerical methods have been proposed, see, for example, [2,8,13,15,16,19], and references therein. In particular, [15,16] consider linear parabolic equations.
In [13] a generalized finite element method (GFEM) was introduced and convergence of optimal order was proven for elliptic multiscale equations. The method builds on ideas from the variational multiscale method [8,10], which is based on a decomposition of the solution space into a (coarse) finite dimensional space and a residual space for the fine scales. The method in [13], often referred to as local orthogonal decomposition, constructs a generalized finite element space where the basis functions contain information from the diffusion coefficient and have support on small vertex patches. With this approach, convergence of optimal order can be proved for an arbitrary positive and bounded diffusion coefficient. Restrictive assumptions such as periodicity of the coefficients or scale separation are not needed. Some recent works [1,6,7,14] show how this method can be applied to boundary value problems, eigenvalue problems, semilinear elliptic equations, and linear wave equations.
In this paper we apply the technique introduced in [13] to parabolic equations with multiscale coefficients. We use the diffusion coefficient to construct a generalized finite element space and for the discretization of the temporal domain we use the backward Euler scheme. Using tools from classical finite element theory for parabolic equations, see, e.g, [11,12,18], and references therein, we prove convergence of optimal order in the L ∞ (L 2 )-norm for linear and semilinear equations under minimal regularity assumptions and nonsmooth initial data. The analysis is completed with numerical examples that support our theoretical findings.
In Sect. 2 we describe the problem formulation and the assumptions needed to achieve sufficient regularity of the solution. Section 3 describes the numerical approximation and presents the resulting GFEM. In Sect. 4 we prove error bounds and in Sect. 5 we extend the results to semilinear parabolic equations. Finally, in Sect. 6 we present some numerical examples.

Problem formulation
We consider the parabolic problem where T > 0 and is a bounded polygonal/polyhedral domain in R d , d ≤ 3. We assume c = c(x), A = A(x), and f = f (x, t). Here we allow both c and A to be multiscale (in space), but independent of the time variable.
We let H 1 ( ) denote the classical Sobolev space with norm v 2 H 1 ( ) = v 2 L 2 ( ) + ∇v 2 L 2 ( ) and V = H 1 0 ( ) the space of functions in H 1 ( ) that vanishes on ∂ . We use H −1 ( ) to denote the dual space to V . Furthermore, we use the notation L p (0, T ; X ) for the Bochner space with finite norm The dependence on the interval [0, T ] and the domain is frequently suppressed and we write, for instance, L 2 (L 2 ) for L 2 (0, T ; L 2 ( )). Finally, we abbreviate the L 2 -norm · := · L 2 ( ) and define |||·||| : To ensure existence, uniqueness, and sufficient regularity, we make the following assumptions on the data.
Due to (A2) this is an inner product and the induced norm c 1/2 · is equivalent to the classical L 2 -norm. We emphasize that throughout this work C denotes a constant that may depend on the bounds α 1 and α 2 (often through the contrast α 2 /α 1 ), the bounds γ 1 and γ 2 , the shape regularity parameter (3.1) of the mesh, the final time T , and the size of the domain , but not on the mesh size parameters nor the derivatives of the coefficients in A or c. The fact that the constant does not depend on the derivatives of A nor c is crucial, since these (if they exist) are large for the problems of interest. This is sometimes also noted as C being independent of the variations of A and c.

Numerical approximation
In this section we describe the local orthogonal decomposition method presented in [13] to define a generalized finite element method for the multiscale problem (2.2). First we introduce some notation. Let {T h } h>0 and {T H } H >h be families of shape regular triangulations of where h K := diam(K ), for K ∈ T h , and H K := diam(K ), for K ∈ T H . We also define H := max K ∈T H H K and h := max K ∈T h h K . Furthermore, we let > 0 denote the shape regularity parameter of the mesh T H ; where B K is the largest ball contained in K . Now define the classical piecewise affine finite element spaces We let N denote the interior nodes of V H and ϕ x the corresponding hat function for x ∈ N , such that span({ϕ x } x∈N ) = V H . We further assume that T h is a refinement of T H , so that V H ⊆ V h . Finally, we also need the finite element mesh T H of to be such that the L 2 -projection P H onto the finite element space V H is stable in H 1 -norm, see, e.g., [3], and the references therein. To discretize in time we introduce the uniform discretization Let U n be the approximation of u(t) at time t = t n and denote f n := f (t n ). Using the notation∂ t U n = (U n − U n−1 )/τ we now formulate the classical backward Euler FEM; find U n ∈ V h such that for n = 1, . . . , N and U 0 ∈ V h is some approximation of u 0 . We also define the operator The convergence of the classical finite element approximation (3.3) depends on D 2 u , where D 2 denotes the second order derivatives. If the diffusion coefficient A oscillates on a scale of we may have D 2 u ∼ −1 , see [17] for a further discussion. The total error is thus typically bounded by u(t n ) − U n ≤ C(τ + (h/ ) 2 ), which is small only if h < .
The purpose of the method described in this paper is to find an approximate solution, let us denote it byÛ for now, in some spaceV ⊂ V h , such that dimV = dim V H , for H > h, and the error U n −Û n ≤C H 2 . Here C is independent of the variations in A and c andÛ n is less expensive to compute than U n . The total error is then the sum of two terms where the first term is the error due to the classical FEM approximation with backward Euler discretization in time. This is small if h (and τ ) is chosen sufficiently small, that is, if h resolves the variations of A. Hence, we think of h > 0 as fixed and appropriately chosen. Our aim is now to analyze the error U n −Û n . We emphasize thatV = V H is not sufficient. The total error would in this case typically be u(t n ) −Û n ∼ (τ + (H/ ) 2 ), which is small only if H < .
The next theorem states some regularity results for (3.3).

5)
and, if f = 0, then where C depends on α 1 , γ 1 , γ 2 and T , but not on the variations of A or c.
and choosing v =∂ t U n we derive and the bounds on c in (A2) we deduce (3.5).
To derive the bounds in (3.6), we define the solution operator E n such that E n v is the solution to (3.3) with f = 0 and initial data v ∈ L 2 . Let {ϕ i } and {λ i } be eigenfunctions and corresponding eigenvalues such that It follows that the eigenvalues {λ i } are real and positive and {ϕ i } are orthogonal with respect to the inner products (·, ·) c and a(·, ·). Furthermore, there is a finite subset of eigenfunctions that spans V h , i.e., span{ϕ i } M i=1 = V h for some M < ∞. With this notation, the solution E n v can be written as The bounds now follows from [18,Lemma 7.3].

Orthogonal decomposition
In this section we describe the orthogonal decomposition which defines the GFEM space denotedV in the discussion above. We refer to [13,14] for details. The GFEM space is defined using only the diffusion coefficient A, that is, the variations in c are not accounted for in the construction of the space. In Sect. 4 we prove that this space indeed is sufficient to obtain convergence of the method.
For the construction of the GFEM space we use the (weighted) Clément interpolation operator introduced in [4], For this interpolation operator the following result is proved [4] whereω K := ∪{K ∈ T H :K ∩ K = ∅} and C depends on the shape regularity .
Using this projection we define the GFEM space, also referred to as the multiscale space, To define a basis for V ms we want to find the projection R f of the nodal basis function ϕ A basis for the multiscale space V ms is thus given by We also introduce the projection R ms : Note that R ms = I − R f . For R ms we have the following lemma, based on the results in [13].

Lemma 3.2 For the projection R ms in (3.10) and v ∈ V h we have the error bound
where C depends on α 1 and , but not on the variations of A or c.
Proof Define the following elliptic auxiliary problem: find z ∈ V h such that In [13, Lemma 3.1] it was proven that the solution to an elliptic equation of the form where C depends on and α 1 , but not on the variations of A. Hence, we have the following bound for z, In particular, if U n is the solution to (3.3), then (3.11) gives The result in Lemma 3.2 should be compared with the error of the classical Ritz which is similar to the result in Lemma 3.2. However, in this case, C depends on the variations of A and the regularity of . This is avoided by using the R ms -projection, since the constant in Lemma 3.2 does not depend on the variations of A or c. Now define the corresponding GFEM to problem (3.3); find U ms n ∈ V ms such that for n = 1, . . . , N . Furthermore, we define the operator A ms : V ms → V ms by In this remark we discuss the possibilities of including time dependency in the coefficients c and A.
(i) It is possible, with a slight modification of the error analysis, to let c = c(x, t) be time dependent with rapid variations in space. However, for simplicity, we shall only study the time independent case here. (ii) We emphasize that the construction of the method in (3.12) depends on the fact that the diffusion coefficient does not depend on time. If we have A = A(x, t), the multiscale space could be updated in each time step. For each t n , we would then define a new Ritz projection R f with A = A(x, t n ) leading to a space V ms n . If the variations in time are slow or periodic, it is also possible reuse the space for several time steps. However, if the variations are fast and non-periodic, then updating the basis may become too expensive which calls for a different approach.

Localization
Since the corrector problems (3.9) are posed in the fine scale space V f they are computationally expensive to solve. Moreover, the correctors φ x generally have global support, which destroys the sparsity of the resulting linear system (3.12). However, as shown in [13], φ x decays exponentially fast away from x. This observation motivates a localization of the corrector problems to smaller patches of coarse elements. Here we use a variant presented in [6], which reduces the required size of the patches.
We first define the notion of patches and their sizes. For all K ∈ T H we define ω k (K ) to be the patch of size k, where and we define R f k := K ∈T H R f K ,k . Hence we can, for each nonnegative integer k, define a localized multiscale space Here the basis is given by The procedure of decomposing V h into the orthogonal spaces V ms and V f together with the localization of V ms to V ms k is referred to as local orthogonal decomposition.
The following lemma follows from Lemma 3.6 in [6].

Lemma 3.4
There exists a constant 0 < μ < 1 that depends on the contrast α 2 /α 1 such that where C depends on α 1 , α 2 , and , but not on the variations of A or c.
The next lemma is a consequence of Theorem 3.7 in [6] and estimates the error due to the localization procedure.
Here C depends on α 1 , α 2 , and , but not on the variations of A or c.
Proof The proof is similar to the proof of Lemma 3.2. Let z ∈ V h be the solution to the elliptic dual problem It follows from Theorem 3.7 in [6] that there exists a constant C depending on α 2 , α 1 , and , such that |||z − R ms We are now ready to formulate the localized version of (3.12) by replacing V ms by V ms k . The localized GFEM formulation reads; find U ms k,n ∈ V ms k such that for n = 1, . . . , N . We also define the operator A ms k : V ms k → V ms k by a localized version of (3.13) We also define the solution operator E ms k,n , such that the solution to (3.16), with f = 0, can be expressed as U ms k,n = E ms k,n U ms k,0 . For this operator we have estimates similar to (3.6). Since the initial data in (3.16) is the projection onto V ms k with respect to the inner product (·, ·) c , we define P ms c,k : to state the next lemma.

Lemma 3.6
For l = 0, 1, and v ∈ L 2 , we have where C depends on the constant α 1 , γ 1 , γ 2 , but not on the variations of A or c.
Proof As in the proof of Theorem 3.1 there exist a finite number of positive eigenvalues It follows that E ms k,n v can be written as

Error analysis
In this section we derive error estimates for the local orthogonal decomposition method introduced in Sect. 3. The bounds of the time derivatives of a parabolic problem with nonsmooth initial data, (c.f. Theorem 3.1), depends on negative powers of t n , which leads to error bounds containing negative powers of t n . These are non-uniform in time, but of optimal order for a fix time t n > 0. The same phenomenon appears in classical finite element analysis for equations with nonsmooth initial data, see [18] and references therein. The error analysis in this section is carried out by only taking the L 2 -norm of U 0 , which allows u 0 ∈ L 2 . Theorem 4.1 Let U n be the solution to (3.3) and U ms k,n the solution to (3.16). Then, for 1 ≤ n ≤ N , where C depends on α 1 , α 2 , γ 1 , γ 2 , , and T , but not on the variations of A or c.
The proof of Theorem 4.1 is divided into several lemmas. To study the error in the homogeneous case, f = 0, we use techniques similar to the classical finite element analysis of problems with nonsmooth initial data, see [18] and the references therein.
Define T h : L 2 → V h and T ms k : With this notation the solution to the parabolic problem (3.3), with f = 0, can be expressed as T h∂t U n + U n = 0, since Similarly, the solution to (3.16), with f = 0, can be expressed as T ms k∂ t U ms k,n + U ms k,n = 0. Note that T h and T ms k are self-adjoint and positive semi-definite with respect to (·, ·) c on L 2 , and T ms k = R ms k T h . Now, let e n = U ms k,n − U n , where e n solves the error equation for n = 1, . . . , N with T ms k e 0 = 0, since The following lemma is a discrete version of [18, Lemma 3.3]. To estimate the last sum we note that, since T ms k is self-adjoint and positive semidefinite,

2(T ms
k∂ t e n , e n ) c = (T ms k∂ t e n , e n ) c + (T ms k e n ,∂ t e n ) c =∂ t (T ms k e n , e n ) c + τ (T ms k∂ t e n ,∂ t e n ) c ≥∂ t (T ms k e n , e n ) c .
so by multiplying the error equation (4.1) by 2ce n we get ∂ t (T ms k e n , e n ) c + 2 c 1/2 e n 2 ≤ 2(T ms k∂ t e n , e n ) c + 2 c 1/2 e n 2 = 2(ρ n , e n ) c .
Multiplying by τ and summing over n gives where we have used that T ms k e 0 = 0. Since the first term is nonnegative we deduce that n j=1 τ c 1/2 e j 2 ≤ n j=1 τ c 1/2 ρ j 2 and (4.2) follows. For n = 1 this also proves (4.3). Note that we have used the bounds on c in (A2) to obtain the result in the L 2 -norm.
Next lemma is a discrete version of a result that can be found in the proof of [ for some > 0. Now define z j = t j e j . Then T ms k∂ t z n + z n = t n ρ n + T ms k e n−1 := η n , n ≥ 1, and, since T ms k z 0 = 0 we conclude from Lemma 4.2 From the definition of η j it follows that where we used 1 2 t j ≤ t j−1 ≤ t j for j ≥ 2. To bound T ms k e n we defineẽ n = n j=1 τ e j andẽ 0 = 0. Multiplying the error equation (4.1) by τ and summing over n gives n j=1 τ T ms k∂ t e j +ẽ n = T ms k∂ tẽn +ẽ n =ρ n , n ≥ 1, whereρ n = n j=1 τρ j and we have used that T ms k e 0 = 0. Note that by definition T ms kẽ 0 = 0. Thus, by Lemma 4.2, we have Hence, since T ms k∂ tẽn = T ms k e n , T ms k e n ≤ ẽ n + ρ n ≤ C max 2≤ j≤n but from (4.3) we deduce z 1 ≤ t 1 ρ 1 , and hence Choosing n * such that max 2≤ j≤n z j = z n * we conclude (4.4).

Lemma 4.4
Assume f = 0 and let U ms k,n be the solution to (3.16) and U n the solution to (3.3). Then, for 1 ≤ n ≤ N , where C depends on α 1 , α 2 , γ 1 , γ 2 , , and T , but not on the variations of A or c.
Proof From Lemma 4.3 we have and from Lemma 4.2 e 1 ≤ C ρ 1 . The rest of the proof is based on estimates for the projection R ms k in Lemma 3.5 and the regularity of the homogeneous equation (3.6). We have where we have used T ms k = R ms k T h and U j ≤ C U 0 .
The next lemma concerns the convergence of the inhomogeneous parabolic problem (2.1) with initial data U 0 = 0. Lemma 4.5 Assume U 0 = 0 and let U ms k,n be the solution to (3.16) and U n the solution to (3.3). Then, for 1 ≤ n ≤ N , where C depends on α 1 , α 2 , γ 1 , γ 2 , , and T , but not on the variations of A or c.
Proof Let U ms k,n − U n = U ms k,n − R ms k U n + R ms k U n − U n =: θ n + ρ n . For ρ n we use Lemma 3.5 to achieve the estimate Using Duhamel's principle we have since θ 0 = 0. Summation by parts now gives Note that ρ 0 = 0. Using Lemma 3.5 and Lemma 3.6 we get where the last sum can be bounded by It remains to bound A h U n . We have A h U n = P h ( f n − c∂ t U n ) and Theorem 3.1 gives which completes the proof.

Proof (of Theorem 4.1)
The result follows from Lemmas 4.4 and 4.5 by rewriting U n = U n,1 + U n,2 , where U n,1 is the solution to the homogeneous problem and U n,2 the solution to the inhomogeneous problem with vanishing initial data.

Remark 4.6
We note that the choice of k and the size of μ determine the rate of the convergence. In general, to achieve optimal order convergence rate, k should be chosen proportional to log(H −1 ), i.e. k = c log(H −1 ). With this choice of k we have U ms k,n − U n ≤ C(1 + log n)H 2 t −1 n .

The semilinear parabolic equation
In this section we discuss how the above techniques can be extended to a semilinear parabolic problem with multiscale diffusion coefficient. In this section we assume, for simplicity, that the coefficient c = 1.

Problem formulation
We are interested in equations of the forṁ where f : R → R is twice continuously differentiable and is a polygonal/polyhedral boundary in R d , for d ≤ 3. For d = 2, 3, f is assumed to fulfill the growth condition where δ = 2 if d = 3 and δ ∈ [1, ∞) if d = 2. Furthermore, we assume that the diffusion A fulfills assumption (A1) and u 0 ∈ V .
Define the ball B R := {v ∈ V : v H 1 ≤ R}. Using Hölder and Sobolev inequalities the following lemma can be proved, see [12].

Lemma 5.2 If f fulfills assumption (5.2) and u, v ∈ B R , then
where C is a constant depending on R.
From (5.1) we derive the variational form; find u(t) ∈ V such that and u(0) = u 0 . For this problem local existence of a solution can be derived given that the initial data u 0 ∈ V , see [12].

Theorem 5.3
Assume that (A1) and (5.2) are satisfied. Then, for u 0 ∈ B R , there exist T 0 = T 0 (R) and C 1 > 0, such that (5.3) has a unique solution u ∈ C(0, T 0 ; V ) and For the Allen-Cahn equation it is possible to find an a priori global bound of u. This means that for any time T there exists R such that if u is a solution then u(t) L ∞ (H 1 ) ≤ R for t ∈ [0, T ]. Thus we can apply the local existence theorem repeatedly to attain global existence, see [12].

Numerical approximation
The assumptions and definitions of the families of triangulations {T h } h>0 and {T H } H >h and the corresponding spaces V H and V h remain the same as in Sect. 3. For the discretization in time we use a uniform time discretization given by and T 0 is given from Theorem 5.3. With these discrete spaces we consider the semiimplicit backward Euler scheme where U n ∈ V h satisfies for n = 1, . . . , N where U 0 ∈ V h is an approximation of u 0 . It is proven in [11] that this scheme satisfies the bound if we choose, for instance, U 0 = P h u 0 , where P h denotes the L 2 -projection onto V h . Note that C in this bound depends on the variations of A.
The following theorem gives some regularity estimates of the solution to (5.5).

Theorem 5.4
Assume that (A1) and (5.2) are satisfied. Then, for U 0 ∈ B R , there exist T 0 = T 0 (R) and C 1 > 0 such that (5.5) has a unique solution U n ∈ V h , for 1 ≤ n ≤ N , and max 1≤n≤N U n H 1 ≤ C 1 R. Moreover, the following bounds hold where C depends on α 1 , T 0 , and R, but not on the variations of A.
Proof We only prove the estimate ∂ t∂t U n ≤ Ct −3/2 n here. The other two follow by similar arguments.
From (5.5) we get Choosing v =∂ t∂t U n in (5.7) gives which gives the bound Using Lemma 5.2 we have for ξ j ∈ (min{U n− j , U n−( j−1) }, max{U n− j , U n−( j−1) }) Note that |ξ 1 − ξ 2 | ≤ |U n−2 − U n−1 | + |U n−1 − U n |. By using Sobolev embeddings we get where we recall the bounds 1 2 t j ≤ t j−1 ≤ t j for j ≥ 2. Multiplying by τ t 4 n in (5.8) and summing over n gives and with ξ j as above, we get where we used Lemma 5.2 and |||∂ t U j ||| ≤ Ct −1 j for j ≥ 1. Multiplying (5.9) with τ t 3 n and summing over n gives where C now depends on t n ≤ T . So we have proved Applying the classical discrete Grönwall's lemma gives which proves ∂ t∂t U n ≤ Ct −3/2 n for n ≥ 3. For n = 2 we proved which completes the proof.
We use the same GFEM space as in Sect. 3, that is, Furthermore, for the completely discrete scheme, we consider the time discretization defined in (5.4) and the linearized backward Euler method thus reads; find U ms k,n ∈ V ms such that U ms k,0 = P ms k U 0 and

Error analysis
For the error analysis we need the following generalized discrete Grönwall lemma, see, e.g., [12].
then there is a constant C depending on B, β 1 , β 2 , and, T , such that, Next lemma states a result for A ms k which is needed in the analysis. A proof of the bound can be found in [12].

Lemma 5.6
The following bound holds where C depends on α 1 , but not on the variations of A.
Theorem 5.7 For given R ≥ 0 and T 0 > 0 let U n be the solution to (5.5) and U ms k,n be the solution to (5.10), such that U n , U ms k,n ∈ B R . Then, for 1 ≤ n ≤ N , where C depends on α 1 , α 2 , , R, and T 0 , but not on the variations of A.
Proof First we define e n = U ms k,n −U n = (U ms k,n − R ms k U n )+(R ms k U n −U n ) = θ n +ρ n . For ρ j we use Lemma 3.5 to prove the bounds For θ n we have To bound θ k,n we first assume n ≥ 2 and use summation by parts for the first part of the sum. Defining n 2 to be the integer part of n/2 we can write −τ n 2 j=1 E ms k,n− j+1 P ms k∂ t ρ j = E ms k,n P ms k ρ 0 − E ms k,n−n 2 P ms k ρ n 2 − τ n 2 j=1∂ t E ms k,n− j+1 P ms k ρ j , and θ n can be rewritten as θ n = E ms k,n P ms k e 0 − E ms k,n−n 2 P ms k ρ n 2 − τ where we note that P ms k e 0 = 0. To estimate these terms we need the following bounds for β 1 see [11]. Using Lemma 3.6 we get and together with Lemmas 3.5 and 5.2 this gives Now consider θ 1 . We can rewrite

Linear parabolic problem
For the linear parabolic problem (2.1) the right hand side is set to f (x, t) = t, which fulfills the assumptions for the required regularity. For simplicity the initial data is set to u 0 = 1. To construct B and D we choose, for each cell in the Cartesian grid, a value from the interval [10 −1 , 10 3 ]. Note that we choose different values for B and D. This procedure gives both B and D rapidly varying features, see Fig. 1. For each value of H the localized GFEM, U ms k,n , and the corresponding P1-FEM, denoted U H,n , are computed. The patch sizes k are chosen such that k ∼ log(H −1 ), that is k = 1, 2, 2, 3, and 4, for the five simulations. When computing U H,n the stiffness matrix is assembled on the fine scale h and then interpolated to the coarser scale. This way we avoid quadrature errors. The convergence results for the two examples are presented in Fig. 2, where the error at the final time t N is plotted against the degrees of freedom |N |. Comparing the plots we can see the predicted quadratic convergence for the localized GFEM. Note that even though the multiscale features of c are not included in the construction of the multiscale space we get convergence without preasymptotic effects, as suggested by the theory. However, as expected, the P1-FEM shows poor convergence on the coarse grids when the coefficients have multiscale features. We clearly see the pre-asymptotic effects when H does not resolve the fine structure of B.

Semilinear parabolic problem
For the semilinear problem we study the Allen-Cahn equation, which has right hand side f (u) = −(u 3 −u) that fulfills the necessary assumptions. We define the initial data to be u 0 (x, y) = x(1 − x)y(1 − y), which is zero on ∂ . The matrix B constructed as in the linear case but with values varying between 10 −3 and 1. Note that, for simplicity, we have c = 1 in both cases.
As in the linear case, we now compute the localized GFEM approximations U ms k,n and the corresponding P1-FEM, U H,n . The patch sizes are chosen to k = 1, 2, 2, 3, and 4, for the five simulations. The convergence results for the two examples are presented in Fig. 3. We can draw the same conclusions as in the linear case. The localized GFEM shows predicted quadratic convergence in both cases, but P1-FEM shows poor convergence on the coarse grids when the coefficients have multiscale features.
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