Multiscale problems are ubiquitous in science and engineering, and the most common manifestation is PDEs with highly oscillatory coefficients. For examples, the physical field equations for modeling composite material, and Darcy’s flows in porous media. Directly solving those problems by universal methods such as finite element method (FEM) is still impractical even with huge development on computational power. Therefore, the general methodology of multiscale computation is utilizing advanced computation architecture, capturing small-scale information while keeping the main workload bearable.
It has been decades since Hou et. al. developed a multiscale finite element method (cf. [12, 13]). Simply put, MsFEM solves a multiscale problem on a coarse mesh while the fine-scale property is revealed in finite element basis, here ”coarse” is comparing to the original characteristic scale, and a great reduction on total freedoms will be achieved. The construction of MsFEM basis is through locally solving a PDE which determined by leading order differential operator. Originally, the boundary condition for those bases is linear to guarantee conformity, later an over-sampling technique together with nonconforming FEM numerical analysis was introduced to weaken the influence of so-called resonance error (see [9, 17]). Recently, a generalized MsFEM performed a further step (), in this method multiscale basis functions are chosen by a two-stage process and has been successfully applied to high contrast flow problems ().
While the blossom in the application, MsFEM remains several numerical analysis results unproved, and this is what our paper mainly concerns. Specifically, an error estimation for MsFEM solution on Dirichlet problem is given in  and states that the error is bounded by
However, a assumption for the homogenized solution seems to be too strong. The proof relies on a boundary corrector , which is intuitive to construct in Dirichlet problem while sightly complicate on Neumann problem. Actually, Chen and Hou did find a proper way in  to define boundary corrector for pure Neumann problem. Nevertheless, their method is highly technical and strict on homogenized solution, also difficult to extend to Robin or nonlinear boundary problems.
Due to the recently developed results for periodic homogenization , we can build the estimation directly from the view of variational form, rather than constructing boundary correct first. This work also motivates our work  on boundary hemivariational inequality— a nonlinear boundary condition which frequently appears in frictional contact modeling. With those estimations prepared, we can now examine the effectiveness of classic MsFEM for various boundary problems clearly and rigorously.
We arrange following section as: in section 2 we first introduce the notations, general settings, and model problems, then review the homogenization theory, highlight on the estimations for later analysis, finally a description for MsFEM is also provided; in section 3 we progressively prove our main results, with a best effort to keep selfconsistency; conclusion remarks are included in section 4.
For simplicity, we consider 2-D problems through the full text. We reserve for a domain with Lipschitz boundary and for spacial dimension (). The Einstein summation convention is adopted, means summing repeated indexes from to . The Sobolev spaces and are defined as usual (see e.g., ) and we abbreviate the norm and seminorm of Sobolev space as and .
2.1 Homogenization Theory
In this subsection, we review the homogenization theory and related estimations for our model problems. Those estimations reveal the creditability of MsFEM on complex boundary conditions, even the crucial boundary corrector in original proof is not available now.
Definition 2.1 (1-periodicity).
A (vector/matrix value) functionis called 1-periodic, if
A 1-periodic (vector/matrix value) function with superscripted means a scaling on as .
The coefficient matrix is symmetric and uniformly elliptic if
Problem A (Mixed Dirichlet-Neumann problem).
Split into two disjointed parts . The problem states:
Its variational form is:
For simplicity, we assume the source term belongs to , and the boundary term .
Problem B (Robin problem).
We consider following problem
with its variational form
We assume to keep the coercivity of bilinear form. In Robin problem, an energy norm can be defined as , which is equivalent to .
Before introducing the hemivariational boundary inequality problem, several notations and definitions will be provided first:
Definition 2.3 (See ).
Let be a locally Lipschitz function. For , the generalized directional derivative of at along the direction , denoted by is defined by
The generalized subdifferential of at , is the nonempty set defined by
In hemivariational inequality problem, we decompose boundary into disjointed parts , and to impose different conditions. Then
Problem C (Boundary hemivariational inequality).
Let be a locally Lipschitz function on , and be the trace operator , the problem states:
which has following hemivariational form:
There exist constants , such that:
Let be the operator norm, there exists that,
Here we use , for abbreviations of and .
The fascinating result of homogenization theory is that can converge as , and this limitation is called homogenized solution which satisfying homogenized equation. One interesting fact is the coefficients corresponded to the equation is not the averages of in periodic cell but elaborate values . Denote and as the completion of smooth 1-periodic function by norm, similarly for .
Definition 2.4 (Correctors and homogenized coefficients).
We denote as correctors for , which satisfy a group of PDEs with periodic boundary condition:
The homogenized coefficients are defined by
We have following theorem:
From the point of computation, weak convergence is inadequate and does not provide quantitative information. Hence an asymptotic expansion is developed and successfully applied to various problems ([1, 2]). Here we only consider first-order expansion .
From the expression of , We can conjecture that twice differentiability of is the least regularity for , and is necessary for the square-integrability of . Actually, those two assumptions will frequently appear in our analysis.
2.2 Multiscale Finite Element Method
Let be a partition of by triangles with circumcircle radius . We need the partition be regular:
Definition 2.8 (Regularity of triangulation family ).
For every element belonging to , denote and as its radii of circumcircle and incircle respectively. The triangulation family is regular if there exists such that for all and for all ,
We only consider Lagrange element space , and take as nodal basis function which vanishes at other nodes except at . Relatively, the multiscale basis can be constructed by following way:
This PDE determines in each element , and a simple observation tells that is locally supported. Here we do not explicitly add on because we want to emphasize MsFEM is a general method, and it is workable even without small periodicity assumption. Let be the finite linear space spanned by , then using MsFEM in problems C, B and A is solving corresponding (hemi)variational form on .
3 Main Results
Let be uniformly elliptic, and as a Lipschitz domain in . Then , we have:
Here we redefine
and the constant only depends on , of , and .
We omit the proof since itself is straight, and we point here the constant in above inequality does not involve geometric description of domain .
Take as triangle with as radius of its incircle. Assume that correctors , and is 1-periodic, symmetric, uniformly elliptic. Let and be the solution of following PDE:
Here . If , then for the error of first-order expansion , we have
here only depends on of and .
Let , and , we will have:
It is obvious that is the homogenized solution of , which immediately leads . According to  p. 6-7, we can construct , such that , and . Then
An integration by parts rule is used in the last line. We obtain , and we are left to estimate . Take , and it is easy to show has bounded derivatives . Here the triangle is simple enough to allow us calculate the specific expression of , and refer  for more general discussions. Denote , and . We will see and on . Then we need to derive the estimation for derivatives of :
Here an important fact that is constant in is considered, and the constant is independent with . The precise value of is , and we have if is small enough comparing to . ∎
By a scaling argument, above lemma is a direct consequence of estimation (refer [16, 14]). However, we must get rid of the dependence on domain in constant , because the elements generated by meshing the domain are not always identical, typically in unstructured grids. This lemma can be naturally extended to 3-D domain.
We need interpolation operator satisfied following property, and one specific example is the local regularization operator in.
If the triangulation is regular, then there exists an interpolation operator and constant such that ,
Here depends on and of .
Take , and with on each . Then we have a lemma:
Assume and . For regular triangulation and interpolation operator satisfying proposition 3.5, we have:
Here is independent with and .
By a direct computation,
Then focus on . Insert and recall the assumption for interpolation , we have
Note on every element boundary, then combine lemma 3.3 and the regularity of ,
The first inequality is established by summing together these parts. For second inequality, by the definition of we have on , then
Here a trace inequality used in the last line. ∎
The Céa’s inequality for hemivariational problem is slightly different (cf. ), and we can obtain
Then we have following theorem:
The proof is clear, and we also note by trace inequality, . The reason we introduce this definition because such estimation is not optimal, and the refined result relies on further development of homogenization theory. Compare theorems 3.8 and 3.7 with first-order expansion method in , a resonance error is inevitably induced because of neglecting the oscillation on inner elements boundary, which triggers considerable subsequent works.
In this paper, we proposed a proof for the convergence rate of MsFEM on mixed Dirichlet-Neumann, Robin and hemivariational inequality boundary problems. The key step is directly utilizing first-order expansion rather than constructing boundary corrector, and this also leads a relaxation on assumptions. Our proof also indicates the originality of resonance error, while whether in those problems this ”annoying” error can be reduced by over-sampling technique is still under considering. Due to the nonlinear setting, the estimation for hemivariational inequality problem is sightly verbose comparing to linear problems, and a refined result can be obtained if further developments from homogenization theory are achieved.
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