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We propose an $hp$-adaptive discontinuous Galerkin finite element method (DGFEM) to approximate the solution of a static crack boundary value problem. The mathematical model describes the behavior of a geometrically linear strain-limiting…
Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is…
The resolution of the Green's function for obtaining the electrostatic potential generated by the charges located in the dielectric layer of a rectangular waveguide requires efficient integration techniques. Due to the characteristics and…
Slender beams are often employed as constituents in engineering materials and structures. Prior experiments on lattices of slender beams have highlighted their complex failure response, where the interplay between buckling and fracture…
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and…
In this thesis we study the lateral electrostatic interaction between a pair of non-identical, moderately charged colloidal particles trapped at an electrolyte interface in the limit of short inter-particle separations. Using a simplified…
We study fully discrete linearized Galerkin finite element approximations to a nonlinear gradient flow, applications of which can be found in many areas. Due to the strong nonlinearity of the equation, existing analyses for implicit schemes…
Particle collisions are the primary mechanism of inter-particle momentum and energy exchange for dense particle-laden flow. Accurate approximation of this collision operator in four-way coupled Euler-Lagrange approaches remains challenging…
A symmetric boundary integral formulation for the transient scattering of acoustic waves off homogeneous and isotropic elastic obstacles is analyzed. Both the acoustic scattered field and the elastodynamic excited field are represented…
We present a hybrid continuous and discontinuous Galerkin spectral element approximation that leverages the advantages of each approach. The continuous Galerkin approximation is used on interior element faces where the equation properties…
A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be…
We present a well-balanced nodal discontinuous Galerkin (DG) scheme for compressible Euler equations with gravity. The DG scheme makes use of discontinuous Lagrange basis functions supported at Gauss-Lobatto-Legendre (GLL) nodes together…
Ab initio calculation of dielectric response with high-accuracy electronic structure methods is a long-standing problem, for which mean-field approaches are widely used and electron correlations are mostly treated via approximated…
A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and…
Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis…
We solve first-kind Fredholm boundary integral equations arising from Helmholtz and Laplace problems on bounded, smooth screens in three-dimensions with either Dirichlet or Neumann conditions. The proposed Galerkin-Bubnov method takes as…
This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many…
The quantum behavior of charge carriers in semiconductor structures is often described in terms of the effective mass Schr\"{o}dinger equation, neglecting the rapid fluctuations of the wave function on the scale of the atomic lattice. For…
We use a concept of weak asymptotic solution for homogeneous as well as non-homogeneous fractional advection dispersion type equations. Using Legendre scaling functions as basis, a numerical method based on Galerkin approximation is…
This paper is concerned with developing accurate and efficient discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in the case of one spatial dimension. The primary…