Local Elliptic Problems Research Articles

A local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations

We introduce a local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations. The proposed method is based on a coarse grid and iteratively improves the solution's accuracy by solving local elliptic problems in refined subdomains. For purely diffusion problems, we already proved that this scheme converges under minimal regularity assumptions (Abdulle and Rosilho de Souza, 2019) [1]. In this paper, we provide an algorithm for the automatic identification of the local elliptic problems' subdomains employing a flux reconstruction strategy. Reliable error estimators are derived for the local adaptive method. Numerical comparisons with a classical nonlocal adaptive algorithm illustrate the efficiency of the method.

A local discontinuous Galerkin gradient discretization method for linear and quasilinear elliptic equations

A local weighted discontinuous Galerkin gradient discretization method for solving elliptic equations is introduced. The local scheme is based on a coarse grid and successively improves the solution solving a sequence of local elliptic problems in high gradient regions. Using the gradient discretization framework we prove convergence of the scheme for linear and quasilinear equations under minimal regularity assumptions. The error due to artificial boundary conditions is also analyzed, shown to be of higher order and shown to depend only locally on the regularity of the solution. Numerical experiments illustrate our theoretical findings and the local method’s accuracy is compared against the non local approach.

Positive solutions to a fractional equation with singular nonlinearity

In this paper, we study the positive solutions to the following singular and non local elliptic problem posed in a bounded and smooth domain Ω⊂RN, N>2s:(Pλ){(−Δ)su=λ(K(x)u−δ+f(u)) in Ωu>0 in Ωu≡0 in RN\\Ω. Here 0<s<1, δ>0, λ>0 and f:R+→R+ is a positive C2 function. K:Ω→R+ is a Hölder continuous function in Ω which behave as dist(x,∂Ω)−β near the boundary with 0≤β<2s.First, for any δ>0 and for λ> small enough, we prove the existence of solutions to (Pλ). Next, for a suitable range of values of δ, we show the existence of an unbounded connected branch of solutions to (Pλ) emanating from the trivial solution at λ=0. For a certain class of nonlinearities f, we derive a global multiplicity result that extends results proved in [2]. To establish the results, we prove new properties which are of independent interest and deal with the behavior and Hölder regularity of solutions to (Pλ).

A high-order multiscale finite-element method for time-domain acoustic-wave modeling

Accurate and efficient wave equation modeling is vital for many applications in such as acoustics, electromagnetics, and seismology. However, solving the wave equation in large-scale and highly heterogeneous models is usually computationally expensive because the computational cost is directly proportional to the number of grids in the model. We develop a novel high-order multiscale finite-element method to reduce the computational cost of time-domain acoustic-wave equation numerical modeling by solving the wave equation on a coarse mesh based on the multiscale finite-element theory. In contrast to existing multiscale finite-element methods that use only first-order multiscale basis functions, our new method constructs high-order multiscale basis functions from local elliptic problems which are closely related to the Gauss–Lobatto–Legendre quadrature points in a coarse element. Essentially, these basis functions are not only determined by the order of Legendre polynomials, but also by local medium properties, and therefore can effectively convey the fine-scale information to the coarse-scale solution with high-order accuracy. Numerical tests show that our method can significantly reduce the computation time while maintain high accuracy for wave equation modeling in highly heterogeneous media by solving the corresponding discrete system only on the coarse mesh with the new high-order multiscale basis functions.

A Renormalization Group Method by Harmonic Extensions and the Classical Dipole Gas

In this paper, we develop a new renormalization group method, which is based on conditional expectations and harmonic extensions, to study functional integrals of small perturbations of Gaussian fields. In this new method, one integrates Gaussian fields inside domains at all scales conditioning on the fields outside these domains, and by the variation principle solves local elliptic problems. It does not rely on an a priori decomposition of the Gaussian covariance. We apply this method to the model of classical dipole gas on the lattice, and show that the scaling limit of the generating function with smooth test functions is the generating function of the renormalized Gaussian free field.

Positive solutions for non local elliptic problems

We establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type for nonlinear problems involving the fractional power of the Dirichlet Laplacian.

A posteriori error estimation for the method of lumped masses applied to second-order hyperbolic problems

We develop a posteriori finite element discretization error estimates for the method of lumped masses applied to the wave equation. In one dimension, we show that the significant part of the spatial finite element error is proportional to a Lobatto polynomial and an error estimate is obtained by solving a set of local elliptic problems. In two dimensions, we show that the dichotomy principle of Babuška and Yu still holds. For even-degree approximations error estimates are computed by solving a set of local elliptic problems and for odd-degree approximations an error estimate is computed using jumps of solution gradients across element boundaries. This study also extends known superconvergence results for elliptic and parabolic problems [L.B. Whalbin, Superconvergence in Galerkin Finite Element Methods, Springer-Verlag, New York, 1995. [26]] to the method of lumped masses for second-order hyperbolic problems.