The method of fundamentalsolutions (MFS)is known as aneffective boundary meshless method. However, the formulation of the MFS results in a dense and extremely ill-conditioned matrix. In this paper we investigate the MFS for solving large-scale problems for the nonhomogeneous modified Helmholtz equation. The key idea is to exploit the exponential decay of the fundamental solution of the modified Helmholtz equation, and consider a sparse or diagonal matrix instead of the original dense matrix. Hence, the homogeneous solution can be obtained efficiently and accurately. A standard two-step solution process which consists of evaluating the particular solution and the homogeneous solution is applied. Polyharmonic spline radial basis functions are employed to evaluate the particular solution. Five numerical examples in irregular domains and a large number of boundary collocation points are presented to show the simplicity and effectiveness of our approach for solving large-scale problems.

We introduce a new variational method for the numerical homogenization of divergence formelliptic, parabolic and hyperbolic equations with arbitrary rough (L∞)coefficients. Our method does not rely on concepts of ergodicity or scale-separation buton compactness properties of the solution space and a new variational approach tohomogenization. The approximation space is generated by an interpolation basis (overscattered points forming a mesh of resolution H) minimizing the L2 norm of thesource terms; its (pre-)computation involves minimizing 𝒪(H−d)quadratic (cell) problems on (super-)localized sub-domains of size 𝒪(H ln(1/H)).The resulting localized linear systems remain sparse and banded. The resultinginterpolation basis functions are biharmonic for d ≤ 3, and polyharmonic ford ≥ 4, forthe operator −div(a∇·) and can be seen as a generalization ofpolyharmonic splines to differential operators with arbitrary rough coefficients. Theaccuracy of the method (𝒪(H)in energy norm and independent from aspect ratios of the mesh formed by the scatteredpoints) is established via the introduction of a new class ofhigher-order Poincaré inequalities. The method bypasses (pre-)computations on the fulldomain and naturally generalizes to time dependent problems, it also provides a naturalsolution to the inverse problem of recovering the solution of a divergence form ellipticequation from a finite number of point measurements.