Related papers: Numerical method based on Galerkin approximation f…
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…
Considering fractional fast diffusion equations on bounded open polyhedral domains in $\mathbb{R}^N$, we give a fully Galerkin approximation of the solutions by $C^0$-piecewise linear finite elements in space and backward Euler…
Although for a number of semilinear stochastic wave equations existence and uniqueness results for corresponding solution processes are known from the literature, these solution processes are typically not explicitly known and numerical…
The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall…
In this paper we investigate the variable coefficient two-sided fractional diffusion, advection, reaction equations on a bounded interval. It is known that the fractional diffusion operator may lose coercivity due to the variable…
This chapter reviews and compares discontinuous Galerkin time-stepping methods for the numerical approximation of second-order ordinary differential equations, particularly those stemming from space finite element discretization of wave…
We propose a discontinuous Galerkin method for convection-subdiffusion equations with a fractional operator of order $\alpha (1<\alpha<2)$ defined through the fractional Laplacian. The fractional operator of order $\alpha$ is expressed as a…
The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline…
We discuss a multiscale Galerkin approximation scheme for a system of coupled quasilinear parabolic equations. These equations arise from the upscaling of a pore scale filtration combustion model under the assumptions of large Damkh\"oler…
We present a full space-time numerical solution of the advection-diffusion equation using a continuous Galerkin finite element method on conforming meshes. The Galerkin/least-square method is employed to ensure stability of the discrete…
Complementarity problems and variational inequalities arise in a wide variety of areas, including machine learning, planning, game theory, and physical simulation. In all of these areas, to handle large-scale problem instances, we need fast…
We propose a procedure for the numerical approximation of invariance equations arising in the moment matching technique associated with reduced-order modeling of high-dimensional dynamical systems. The Galerkin residual method is employed…
We use a finite element approach based on Galerkin method to obtain approximate steady state solutions of the thermistor problem with temperature dependent electrical conductivity.
In this paper, we develop a fully discrete Galerkin method for solving initial value fractional integro-differential equations(FIDEs). We consider Generalized Jacobi polynomials(GJPs) with indexes corresponding to the number of homogeneous…
In this paper, a new type of the discrete fractional Gr{\"o}nwall inequality is developed, which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdiffusion equation. Based on…
A semidiscrete Galerkin finite element method applied to time-fractional diffusion equations with time-space dependent diffusivity on bounded convex spatial domains will be studied. The main focus is on achieving optimal error results with…
We introduce a very weak space-time variational formulation for the wave equation, prove its well-posedness (even in the case of minimal regularity) and optimal inf-sup stability. Then, we introduce a tensor product-style space-time…
Using the Galerkin method, we obtain the unique existence of the weak solution to a time fractional wave problem, and establish some regularity estimates which reveal the singularity structure of the weak solution in time.
Subsurface flows are commonly modeled by advection-diffusion equations. Insufficient measurements or uncertain material procurement may be accounted for by random coefficients. To represent, for example, transitions in heterogeneous media,…
Near-optimal computational complexity of an adaptive stochastic Galerkin method with independently refined spatial meshes for elliptic partial differential equations is shown. The method takes advantage of multilevel structure in expansions…