Multiscale finite element methods

Take a symmetric uniformly elliptic coefficient Aε on a shape-regular simplicial coarse mesh. For each element K and each local vertex, solve −div(Aε∇φiK)=0 inside K with the corresponding affine nodal hat function prescribed on its boundary. Adjacent elements use matching affine traces on their common face. Assembling the local functions therefore gives continuous H¹ trial functions, with boundary nodes treated consistently with the global Dirichlet condition. The sum of the local basis functions is one because its boundary value is one and the constant function solves the local equation uniquely.

Insert this conforming space into the original weak bilinear form, integrating the oscillatory coefficient against the multiscale basis gradients. Coercivity gives a well-posed discrete system; Galerkin orthogonality and Céa's lemma compare its energy error to the best approximation in this particular space. Those statements do not estimate that best approximation automatically. The local PDEs and the bilinear integrals must also be solved accurately. An exact mathematical basis does not justify using an unresolved point sample for a rapidly oscillating stiffness integral.

Let K be an interval with endpoints left and right. A local harmonic function has constant flux, so the right-endpoint basis is the accumulated resistance from the left endpoint divided by the total interval resistance RK. Its complement gives the left-endpoint basis. Differentiating shows the basis gradient is inversely proportional to the local conductivity. This redistributes strain into the low-conductivity phase exactly as the cell corrector does, while matching the prescribed coarse nodal values.

The resulting two-by-two stiffness matrix is one divided by RK times the usual matrix with diagonal entries one and off-diagonal entries minus one. Its energy for a nodal jump c is c²/(2RK). In contrast, a conventional affine basis uses a uniform gradient and gives a stiffness involving the arithmetic conductivity integral divided by the square of the element length. The two coincide for constant coefficients but can differ substantially in layered media. Exact local integration therefore changes the response even though both spaces have the same nodal degrees of freedom.

When periodic resistance has a bounded mean-zero primitive, integrating over an interval of length H leaves an endpoint remainder of order ε. Thus RK equals H times mean resistance plus an order-ε phase-dependent remainder. The relative discrepancy can be of order ε/H. This elementary estimate explains why a local window containing few cells can misrepresent the bulk response. Exact alignment with whole periods can cancel that remainder in one dimension, but such cancellation is a special geometry rather than a general numerical guarantee.

In several dimensions, artificial local boundary conditions can create resonance errors between the coarse element and the oscillations. Oversampling computes local functions on enlarged patches to reduce sensitivity to those artificial boundaries, but the resulting space and conformity treatment must be specified. No universal rate follows merely from using that name. A complete error argument balances coarse approximation, microscopic boundary effects, local discretization, quadrature, and any oversampling parameters. Homogenization suggests effective behavior; numerical analysis must still show that the chosen algorithm actually captures it.