Local discontinuous Galerkin method for a third‐order singularly perturbed problem of reaction–diffusion type

Abstract

The local discontinuous Galerkin (LDG) method is studied for a third-order singularly perturbed problem of the convection-diffusion type. Based on a regularity assumption for the exact solution, we prove almost O(N −(k+1/2) ) (up to a logarithmic factor) energy-norm convergence uniformly in the perturbation parameter. Here, k ≥ 0 is the maximum degree of piecewise polynomials used in discrete space, and N is the number of mesh elements. The results are valid for the three types of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin type, and Bakhvalov-type. Numerical experiments are conducted to test the theoretical results.

Local discontinuous Galerkin method on graded meshes for a third‐order singularly perturbed problem

“…Then the LDG solution determined by ( 13) exists uniquely, because there exists only zero solution for the corresponding homogeneous linear equations of (13), that is, 𝑈 = 0 if one sets 𝑓 = 0 and 𝝌 = (𝑈, −𝑄, 𝑃) in (13). Besides, 𝑃 = 𝑄 = 0 if one sets 𝝌 = (0, 𝑃, 0) and 𝝌 = (0, 0, 𝑄) in ( 13) respectively.…”

“…The details are thus omitted. be the numerical solution of the LDG scheme (13) on the graded mesh (29). There exists a constant 𝐶 > 0 independent of 𝜀 and 𝑁 such that…”

“…This result was later extended to several layer‐adapted meshes including Bakhvalov‐Shishkin type mesh and Bakhvalov‐type mesh [12]. Recently, we discuss the third‐order singularly perturbed problem (1) in [13], where a quasi‐optimal uniform convergence of the LDG method is derived on several layer‐adapted meshes using the mesh‐generating functions.…”