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We investigate high-order Convolution Quadratures methods for the solution of the wave equation in unbounded domains in two dimensions that rely on Nystr\"om discretizations for the solution of the ensemble of associated Laplace domain…
We derive the solvability and regularity of the Dirichlet problem for fully non-linear elliptic equations possibly with degenerate right-hand side on Hermitian manifolds, through establishing a quantitative version of boundary estimate…
In this paper, we revisit the nonoverlapping domain decomposition methods for solving elliptic problems with high contrast coefficients. Some interesting results are discovered. We find that the Dirichlet-Neumann algorithm and Robin-Robin…
The Dirichlet-Neumann method is a common domain decomposition method for nonoverlapping domain decomposition and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are…
The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form $r^\lambda v$ is reduced to…
We solve the Dirichlet problem for $k$-Hessian equations on compact complex manifolds with boundary, given the existence of a subsolution. Our method is based on a second order a priori estimate of the solution on the boundary with a…
We present and analyze a non-conforming domain decomposition approximation for a hypersingular operator governed by the Helmholtz equation in three dimensions. This operator appears when considering the corresponding Neumann problem in…
The solution of partial differential equations (PDEs) on complex domains often presents a significant computational challenge by requiring the generation of fitted meshes. The Diffuse Domain Method (DDM) is an alternative which reformulates…
Wave propagation and acoustic scattering problems require vast computational resources to be solved accurately at high frequencies. Asymptotic methods can make this cost potentially frequency independent by explicitly extracting the…
In this paper we present an overview of recent progress on the development and analysis of domain decomposition preconditioners for discretised Helmholtz problems, where the preconditioner is constructed from the corresponding problem with…
We consider the inverse problem of determining coefficients appearing in semilinear elliptic equations stated on Riemannian manifolds with boundary given the knowledge of the associated Dirichlet-to-Neumann map. We begin with a negative…
We study the Dirichlet problem for a class of curvature equations arising from conformal geometry on Riemannian manifolds $(M^n, g)$ with boundary where $n \geq 3$. We prove there exists a unique solution using the continuity method which…
We study spaces of modelled distributions with singular behaviour near the boundary of a domain that, in the context of the theory of regularity structures, allow one to give robust solution theories for singular stochastic PDEs with…
Simulation of physical systems is one of the most promising use cases of future digital quantum computers. In this work we systematically analyze the quantum circuit complexities of block encoding the discretized elliptic operators that…
The Stokes problem with non-homogeneous Dirichlet boundary condition is solved numerically using conforming discretizations and an approximation of the boundary datum in the corresponding trace space. Optimal discretization error estimates…
The orthogonality of Hilbert spaces whose elements can be represented as simple and double layer potentials is determined. Conditions of well-posed solvability of integral equations for the sum of simple and double layer potentials…
A method is presented for the analytical evaluation of the singular and near-singular integrals arising in the Boundary Element Method solution of the Helmholtz equation. An error analysis is presented for the numerical evaluation of such…
In this article, a parameter-uniform numerical method is presented to solve one-dimensional singularly perturbed parabolic convection-diffusion turning point problem exhibiting two exponential boundary layers. We study the asymptotic…
Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of Boundary Integral Equations (BIE), this approach affords solutions of Navier problems via the simpler…
This paper presents a regularization technique for the high order efficient numerical evaluation of nearly singular, principal-value, and finite-part Cauchy-type integral operators. By relying on the Cauchy formula, the Cauchy-Goursat…