Virtual Element Methods for Elliptic Interface Problems in Three Dimensions

math paper

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We consider numerical methods for solving the following interface problems

$$\nabla \cdot (\beta \nabla u) = f ...$$

(Min: please add details)

Project 1: VEM for elliptic interface problems
* $\checkmark$ VEM for elliptic equations
* $\checkmark$ Interface meshes in 3-D
* $\square$ VEM for interface problems: simple geometry
- $\checkmark$ function continuous, flux continuous
- $\checkmark$ function continuous, flux jump
- $?$ function jump, flux jump
* $\square$ Adaptive Interface meshes in 3-D
- refinement using octtree
- geometry information on the interface

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Project 2: Improved Accuracy
* $\square$ Isoparametric Interface VEM in 2-D and 3-D
* $\square$ High order Interface VEM in 2-D and 3-D

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Project 3: Application

Discussion

Function Jump
(Long) I want to bring up the discussion on the way to deal with the jump of function values. Now the numerical results shows that:
1. if the function is continuous, no matter how big the flux jump is, the $L^{\infty}$ norm is always second order.
2. if the function is discontinuous, the order is reduced to 1.6 ~ 1.8. As $h\to \infty$, the order is approaching $2$.
3. The maximum norm for $\Omega_{-}$ and $\Omega_{+}$ is different. One is stricktly second order. Another is small but not second order.
(Min, please add information on your test)

Mathematically we introduce a function $w$ to eliminate the function value jump. But does the approximation depends on the choice of $w$? On one hand, it seems that the solution $p = u - w$ depends on the $w$ if we think about the algebraic equation. The matrix $A$ is always the same. The vector $b$ for $f$ and flux jump $q_1$ is the same. Then different $w$ will give different right hand side and consequently different solution. The solution is only modified in $\Omega_{-}$. Can we prove or numerical test the dependence of the choice of the extension?

Here is a simple test. Chose 1-D intervale (0,1). Consider the interface problem with interface located at $1/4$ and $3/4$. The exact solution is taken as piecewise constant function. It is zero outside and one inside, i.e., $(1/4, 3/4)$. The $\beta$ can be simply 1 first and the flux jump is set as zero. The function jump is $1$ (or $-1$). Then form the matrix and test different $w$.
Min: Can you code this simple test?