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An exponential integration generalized multiscale finite element method for parabolic problems
(2023-04-15)
We consider linear and semilinear parabolic problems posed in high-contrast multiscale media in two dimensions.
The presence of high-contrast multiscale media adversely affects the accuracy, stability, and overall efficiency ...
A Deep Double Ritz Method (D2RM) for solving Partial Differential Equations using Neural Networks
(2023-02-15)
Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min–max) problem over the ...
Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of phi-functions of matrices via quadrature
(2023-02-04)
In this article, we propose an algorithm for approximating the action of $\varphi$-functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with ...
Combining DPG in space with DPG time-marching scheme for the transient advection–reaction equation
(2022-12-01)
In this article, we present a general methodology to combine the Discontinuous Petrov-Galerkin (DPG) method in space and time in the context of methods of lines for transient advection-reaction problems. We rst introduce ...
Exploiting the Kronecker product structure of φ−functions in exponential integrators
(2022-05-15)
Exponential time integrators are well-established discretization methods for time semilinear systems of ordinary differential equations. These methods use (Formula presented.) functions, which are matrix functions related ...
Error representation of the time-marching DPG scheme
(2022-03-01)
In this article, we introduce an error representation function to perform adaptivity in time of the recently developed time-marching Discontinuous Petrov–Galerkin (DPG) scheme. We first provide an analytical expression for ...
The DPG Method for the Convection-Reaction Problem, Revisited
(2022-01-01)
We study both conforming and non-conforming versions of the practical DPG method for the convection-reaction problem. We determine that the most common approach for DPG stability analysis - construction of a local Fortin ...
Equivalence between the DPG method and the Exponential Integrators for linear parabolic problems
(2020-11)
The Discontinuous Petrov-Galerkin (DPG) method and the exponential integrators are two well established numerical methods for solving Partial Di fferential Equations (PDEs) and sti ff systems of Ordinary Di fferential ...
A DPG-based time-marching scheme for linear hyperbolic problems
(2020-11)
The Discontinuous Petrov-Galerkin (DPG) method is a widely employed discretization method for Partial Di fferential Equations (PDEs). In a recent work, we applied the DPG method with optimal test functions for the time ...
Explicit-in-Time Variational Formulations for Goal-Oriented Adaptivity
(2019-10)
Goal-Oriented Adaptivity (GOA) is a powerful tool to accurately approximate physically relevant features of the solution of Partial Differential Equations (PDEs). It delivers optimal grids to solve challenging engineering ...