We present a high-order wavelet Galerkin method to solve the Helmholtz equation in an electromagnetic scattering problem such that the condition numbers are uniformly bounded. Through self-contained simple proofs, we prove that tensor products of biorthogonal wavelets in $L^2([0,1])$, after a scaling, form Riesz bases in the Sobolev space $H^1((0,1)^d)$. This paper also provides many optimized wavelets in Sobolev spaces for the Helmholtz equation.

This paper introduces a new notion of interpolating refinable functions and ns-step interpolatory subdivision schemes. Then then characterize their convergence in terms of their masks. Moreover, a notion of quasi-stationary subdivision schemes is introduced and their convergence is provided.

This paper develops sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem such that the hybrid FDMs utilize a 9-point compact stencil at any interior regular points of the grid and a 13-point stencil at irregular points near interface. For the elliptic problem without interface, our compact 9-point FDM has the M-matrix property for any mesh size and consequently, satisfies the discrete maximum principle, which guarantees the theoretical sixth-order convergence. Numerically, we always use function values to approximate all required high order (partial) derivatives in our hybrid FDMs without losing accuracy. Our proposed FDMs are independent of the choice representing the interface and are also applicable if the jump conditions on interfaces only depend on the geometry (e.g., curvature) of the interface. Our numerical experiments confirm the sixth-order convergence in the $l_{\infty}$ norm of the proposed hybrid FDMs for the elliptic interface problem.

For a general matrix, we show that there is only one meaningful way of defining a vector subdivision scheme. Motivated by vector cascade algorithms and recent study on Hermite subdivision schemes, we shall define a vector subdivision scheme for any arbitrary matrix mask and then we prove that the convergence of the newly defined vector subdivision scheme is equivalent to the convergence of its associated vector cascade algorithm. We also study convergence rates of vector subdivision schemes.

Hermite subdivision schemes iteratively compute limiting functions and t\ heir consecutive derivatives. This paper introduces multivariate generalized Hermite subdivision schemes incl\ uding known subdivision schemes as special cases and introduces new schemes such as Birkhoff or Lagrange subdiv\ ision schemes. We characterize their convergence, smoothness and underlying matrix masks. We also\ introduce a notion of linear-phase moments of matrix masks for achieving polyn\ omial-interpolation property. We constructively prove that there always exist c\ onvergent arbitrarily smooth multivariate generalized Hermite subdivision schem\ es with linear-phase moments such that their basis vector functions are splines\ and have linearly independent shifts.

Iteratively computing a limiting function and its consecutive derivatives, a Hermite subdivision scheme is of particular interest and importance in CAGD for generating smooth subdivision curves and in numerical PDEs for constructing Hermite wavelets. This paper characterizes the convergence and smoothness of a Hermite subdivision scheme, provides simple factorization of Hermite masks through the normal form of matrix-valued masks, and presents an algorithm for constructing all masks for Hermite subdivision schemes.

Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. In this paper, we present a sixth-order compact finite difference method for 2D Helmholtz equations with singular sources, which can also handle any possible combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several state-of-the-art finite difference schemes, particularly in the critical pre-asymptotic region where $\ka h$ is near $1$ with $\ka$ being the wavenumber and $h$ the mesh size.

Factorization of matrices of Laurent polynomials plays an important role in mathematics and engineering such as wavelet frame construction and filter bank design. Motivated by the recent development of quasi-tight framelets, we study and characterize generalized spectral factorizations with symmetry for 2-by-2 matrices of Laurent polynomials. Applying our result on generalized matrix spectral factorization, we establish a necessary and sufficient condition for the existence of symmetric quasi-tight framelets with two generators. The proofs of all our main results are constructive and therefore, one can use them as construction algorithms. We provide several examples to illustrate our theoretical results on generalized matrix spectral factorization and quasi-tight framelets with symmetry.

Though the elliptic interface problem along a smooth non-intersecting interface has been extensively studied, there are few papers addressing the elliptic interface problem with intersecting interfaces. For the elliptic cross-interface problem with 4 discontinuous constant coefficients, we obtain compact 9-point finite difference methods FDMs) of accuracy order 4 using uniform Cartesian meshes. Moreover, for the special case that the interface intersecting point is a grid point, such compact FDMs can even reach the accuracy order 6 and we prove their theoretical convergence rate of order 6 using the discrete maximum principle. Our numerical experiments demonstrate the fifth (for the general case) and sixth (for the special case) accuracy orders of our proposed FDMs. For general cross-interfaces, to achieve the M-matrix property, we derive compact FDMs of the accuracy order 3, leading to the theoretical third order convergence of such proposed compact FDMs using the discrete maximum principle for the general elliptic cross-interface problem.

Numerically solving the 2D Helmholtz equation is widely known to be \ very difficult largely due to its highly oscillatory solution. A very fine mesh\ size is necessary to deal with a large wavenumber leading to a severely ill-co\ nditioned huge coefficient matrix. To understand and tackle such challenges, it\ is crucial to analyze how the solution of the 2D Helmholtz equation depends on\ boundary and source data for large wavenumbers. In this paper, we analyze and\ derive several new sharp wavenumber-explicit stability bounds for the 2D Helmh\ oltz equation with inhomogeneous mixed boundary conditions: Dirichlet, Neumann,\ and impedance.

We introduce bivariate dual sqrt{2}-subdivision schemes using 1D stencil\ s. Such subdivision schemes enjoy the quasi-interpolating property (i.e., they \ interpolate bivariate polynomials of high orders) and are intrinsically linked \ to both 1D primal interpolating subdivision schemes and 1D masks having linear-\ phase moments. Using only 1D stencils, such subdivision schemes can be straight\ forwardly implemented on any quadrilateral meshes and there is no need to desig\ n special subdivision rules near extraordinary or boundary vertices.

The elliptic interface problems with discontinuous and high-contrast coefficients have many applications but are very challenging to numerically solve, because their solutions often have low regularity or even discontinuity (which result in huge highly ill-conditioned linear systems). Thus, it is highly desired to construct high order schemes to solve the elliptic interface problems with discontinuous and high-contrast coefficients. To solve such elliptic interface problem with a smooth interface curve, we propose a fourth order compact finite difference method for computing the solution and its gradient by using uniform meshes. We showe that our proposed FDM numerically satisfies the discrete maximum principle when the mesh size is small and then we use it to prove their convergence of four order.

Using newly modified cubic B-splines with accuracy order 4, we proposed a numerical scheme for solving the 1D and 2D hyperbolic telegraph equation using a differential quadrature method. Our modified cubic B-splines retain the tridiagonal structure and achieve the fourth order convergence rate. The solution of the associated ODEs is advanced in the time domain by the SSPRK scheme.