High accuracy finite difference approximation to solutions of elliptic partial differential equations

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Partial Differential Equations

Background:

• Solving linear elliptic partial differential equations (PDEs) with general linear boundary conditions is crucial in various scientific and engineering fields.
• Existing numerical methods may face limitations in achieving high-order accuracy for complex PDEs, especially those with variable coefficients.

Purpose of the Study:

• To introduce a flexible finite difference method for obtaining high-order accurate approximate solutions to linear elliptic PDEs.
• To demonstrate the method's applicability to equations with general linear boundary conditions and variable coefficients.

Main Methods:

• A finite difference method is developed, determining unknown function values at mesh points by solving a system of finite difference equations L(h)U = I(h)G.
• Coefficients for the difference operators L(h) and I(h) are determined through local calculations to ensure exact approximation on a specific function space.
• The resulting system of equations is solved using standard techniques to obtain approximations at interior mesh points.

Main Results:

• The method achieves high-order accuracy, generalizing known results for the Poisson equation to variable coefficient equations of the form -div(p grad[u]) + Fu = G.
• The approximation order of O(h^6) is demonstrated for smooth solutions of these more general equations.
• The flexibility of the method allows for its application to non-elliptic problems as well.

Conclusions:

• The proposed finite difference method offers a robust and accurate approach for solving a wide range of linear elliptic PDEs.
• Its ability to achieve high-order accuracy, even for variable coefficient problems, makes it a valuable tool in computational science.
• The method's adaptability suggests potential for broader applications beyond elliptic equations.