We propose a multiscale model reduction method for partial differential equations. The main purpose of this method is to derive an effective equation for multiscale problems without scale separation. An essential ingredient of our method is to decompose the harmonic coordinates into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is in H2, and can be approximated by a regular coarse mesh. When the multiscale problem has scale separation and a periodic structure, our method recovers the traditional homogenized equation. Furthermore, we provide error analysis for our method and show that the solution to the effective equation is close to the original multiscale solution in the H1 norm. Numerical results are presented to demonstrate the accuracy and robustness of the proposed method for several multiscale problems without scale separation, including a problem with a high contrast coefficient.

We study the stability properties of explicit marching schemes for second-kind Volterra integral equations that arise when solving boundary value problems for the heat equation by means of potential theory. It is well known that explicit finite-difference or finite-element schemes for the heat equation are stable only if the time step 􏰃t is of the order O(􏰃x2), where 􏰃x is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions d ≥ 1, we show that the simplest Volterra marching scheme, i.e., the forward Euler scheme, is unconditionally stable. Our proof is based on an explicit spectral radius bound of the marching matrix, leading to an estimate that an L2-norm of the solution to the integral equation is bounded by cd T d /2 times the norm of the right-hand side. For the Robin problem on the half-space in any dimension, with constant Robin (heat transfer) coefficient κ, we exhibit a constant C such that the forward Euler scheme is stable if 􏰃t < C /κ 2 , independent of any spatial discretization. This relies on new lower bounds on the spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally, we show that the forward Euler scheme is unconditionally stable for the Dirichlet problem on any smooth convex domain in any dimension, in the L∞-norm.

Two fundamental difficulties are encountered in the numerical evaluation of time- dependent layer potentials. One is the quadratic cost of history dependence, which has been successfully addressed by splitting the potentials into two parts—a local part that contains the most recent contributions and a history part that contains the con- tributions from all earlier times. The history part is smooth, easily discretized using high-order quadratures, and straightforward to compute using a variety of fast algo- rithms. The local part, however, involves complicated singularities in the underlying Green’s function. Existing methods, based on exchanging the order of integration in space and time, are able to achieve high-order accuracy, but are limited to the case of stationary boundaries. Here, we present a new quadrature method that leaves the order of integration unchanged, making use of a change of variables that converts the singular integrals with respect to time into smooth ones. We have also derived asymptotic formulas for the local part that lead to fast and accurate hybrid schemes, extending earlier work for scalar heat potentials and applicable to moving boundaries. The performance of the overall scheme is demonstrated via numerical examples.

We consider finite difference schemes based on the impulse density variable. We show that the original velocity—impulse density formulation of Oseledets is marginally ill-posed for the inviscid flow, and this has the consequence that some ordinarily stable numerical methods in other formulations become unstable in the velocity—impulse density formulation. We present numerical evidence of this instability. We then discuss the construction of stable finite difference schemes by requiring that at the numerical level the nonlinear terms be convertible to similar terms in the primitive variable formulation. Finally we give a simplified velocity—impulse density formulation which is free of these complications and yet retains the nice features of the original velocity—impulse density formulation with regard to the treatment of boundary. We present numerical results on this simplified formulation for the driven cavity flow on both the staggered and non-staggered grids.

Photonic graphene, a photonic crystal with honeycomb structures, has been intensively studied in both theoretical and applied fields. Similar to graphene which admits Dirac Fermions and topological edge states, photonic graphene supports novel and subtle propagating modes (edge modes) of electromagnetic waves. These modes have wide applications in many optical systems. In this paper, we propose a novel gradient recovery method based on Bloch theory for the computation of topological edge modes in the honeycomb structure. Compared to standard finite element methods, this method provides higher order accuracy with the help of gradient recovery technique. This high order accuracy is highly desired for constructing the propagating electromagnetic modes in applications. We analyze the accuracy and prove the superconvergence of this method. Numerical examples are presented to show the efficiency by computing the edge mode for the P-symmetry and C-symmetry breaking cases in honeycomb structures.