Related papers: Coercivity, essential norms, and the Galerkin meth…
We present new second-kind integral-equation formulations of the interior and exterior Dirichlet problems for Laplace's equation. The operators in these formulations are both continuous and coercive on general Lipschitz domains in…
This paper is concerned with solving the Helmholtz exterior Dirichlet and Neumann problems with large wavenumber $k$ and smooth obstacles using the standard second-kind boundary integral equations. We consider Galerkin and collocation…
We say that $\Gamma$, the boundary of a bounded Lipschitz domain, is locally dilation invariant if, at each $x\in \Gamma$, $\Gamma$ is either locally $C^1$ or locally coincides (in some coordinate system centred at $x$) with a Lipschitz…
A new space-time discontinuous Galerkin (dG) method utilising special Trefftz polynomial basis functions is proposed and fully analysed for the scalar wave equation in a second order formulation. The dG method considered is motivated by the…
We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these…
In terms of layer potential methods, this paper is devoted to study the $L^2$ boundary value problems for nonhomogeneous elliptic operators with rapidly oscillating coefficients in a periodic setting. Under a low regularity assumption on…
We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz single- and double-layer boundary-integral operators as mappings from $L^2(\partial \Omega)\rightarrow H^1(\partial \Omega)$ (where $\partial\Omega$ is the…
This article concerns the weak Galerkin mixed finite element method (WG-MFEM) for second order elliptic equations on 2D domains with curved boundary. The Neumann boundary condition is considered since it becomes the essential boundary…
We develop and analyze a local discontinuous Galerkin (LDG) method for solving integral fractional Laplacian problems on bounded Lipschitz domains. The method is based on a three-field mixed formulation involving the primal variable, its…
This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces,…
The Dual Characteristic-Galerkin method (DCGM) is conservative, precise and experimentally positive. We present the method and prove convergence and $L^2$-stability in the case of Neumann boundary conditions. In a 2D numerical finite…
Let $\Omega$ be a bounded domain in R d with Lipschitz boundary $\Gamma$. We define the Dirichlet-to-Neumann operator N on L 2 ($\Gamma$) associated with a second order elliptic operator A = -- d k,j=1 $\partial$ k (c kl $\partial$ l) + d…
We present and analyze a discontinuous Galerkin method for the numerical solution of a class of second-order linear mixed-type partial differential equations, i.e. equations that change their nature from elliptic to hyperbolic through the…
This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. It is…
In this paper, a second-order linearized discontinuous Galerkin method on general meshes, which treats the backward differentiation formula of order two (BDF2) and Crank-Nicolson schemes as special cases, is proposed for solving the…
We present a general numerical method for computing guaranteed two-sided bounds for principal eigenvalues of symmetric linear elliptic differential operators. The approach is based on the Galerkin method, on the method of a priori-a…
We use the method of layer potentials to study the $R_2$ Regularity problem and the $D_2$ Dirichlet problem for second order elliptic equations of the form $\mathcal{L}u=0$, with lower order coefficients, in bounded Lipschitz domains. For…
We develop novel first-kind boundary integral equations for Euclidean Dirac operators in 3D Lipschitz domains comprising square-integrable potentials and involving only weakly singular kernels. Generalized Garding inequalities are derived…
A family of interior penalty $hp$-discontinuous Galerkin methods is developed and analyzed for the numerical solution of the quasilinear elliptic equation $-\nabla{} \cdot (\mathbf{A}(\nabla{u}) \nabla{u} = f$ posed on the open bounded…
Let $\Omega$ be a Lipschitz domain in $\mathbb R^n$ $n\geq 2,$ and $L=\mbox{div} (A\nabla\cdot)$ be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in…