The finite element method is employed to derive precise bounds for the eigenvalues of differential operators. Historically, establishing upper bounds for eigenvalues has been successfully achieved. However, providing lower bounds for eigenvalues remains a significant challenge. In contrast to traditional eigenvalue analysis, which predominantly concentrates on efficient algorithms for eigenvalue approximations, recent advancements have fostered the development of novel theories and computational techniques to ascertain both rigorous lower and upper bounds for these eigenvalues.
The Lehmann-Goerisch theorem plays a pivotal role in obtaining high-precision bounds for eigenvalues using conforming approximate eigenfunctions, such as those derived from conforming finite element methods (FEMs).
The traditional error analysis pertaining to the finite element method primarily focuses on qualitative aspects, such as the convergence rates of numerical schemes. However, in numerous scenarios, such as when seeking a computer-assisted mathematical proof, it is imperative to obtain precise values or upper bounds of approximation errors. Addressing this necessitates confronting the subsequent challenges, which are intrinsically linked to the eigenvalue problem of differential operators:
- Providing not only boundedness but also explicit values for various error constants within the error analysis.
- Addressing the singularity present in partial differential equation (PDE) solutions.
Verified computation seeks to address numerous challenges in order to deliver rigorous computational results, which can even be incorporated into mathematical proofs.
To achieve this, interval arithmetic is employed to handle the rounding errors arising from computations with floating-point numbers. Additionally, the development of novel computational algorithms and error analysis theories is essential to assess various errors, for example, the truncation error and the function approximation error.
A significant area within verified computation involves exploring the existence and uniqueness of solutions to non-linear partial differential equations, such as the Navier-Stokes equation.
The four-probe method serves as an efficient approach to gauging the resistivity of semiconductor materials. The underlying mathematical model is predicated upon the subsequent Poisson boundary value problem:
\( -\Delta u = f \text{ in } \Omega, {\partial{u}}/{{\partial n}} =0 \text{ on } \partial \Omega . \)
By resolving the aforementioned problem within a 3D domain and addressing the singularity in the solution, one can derive the resistivity. For a more comprehensive discussion, please refer to the provided details (URL).
- This is research is supported by JKA RING!RING! Project.