# High-Order Verified Solutions of the 3D Laplace Equation

### Abstract

For many practical problems, numerical methods to solve partial differential equations
(PDEs) are required. Conventional finite element or finite difference codes have a
difficulty to obtain precise solutions because of the need for an exceedingly fine mesh
which leads to often prohibitive CPU time. While conventional methods exhibit such a
difficulty, some practical problems even require solutions guaranteed. The Laplace
equation is one of the important PDEs in physics and engineering, describing the
phenomenology of electrostatics and magnetostatics among others, and various problems
for the Laplace equation require highly precise and verified solutions.
We present an alternative approach based on high-order quadrature and a high-order finite
element method utilizing Taylor model methods. An n-th order Taylor model of a
multivariate function f consists of an $n$-th order multivariate Taylor polynomial,
representing a high order approximation of the underlying function f, and a remainder
error bound interval for verification, width of which scales in (n+1)-st order.

The solution of the Laplace equation in space is first represented as a Helmholtz integral
over the two-dimensional surface. The latter is executed by evaluating the kernel of the
integral as a Taylor model of both the two surface variables and the three volume
variables inside the cell of interest. Finally, the integration over the surface variables
is executed, resulting in a local Taylor model of the solution within one cell.
Examples of the method will be given, demonstrating achieved accuracy with verification.

S. Manikonda, M. Berz, K. Makino,
* Transactions on Computers * **11,4** (2005) 1604-1610

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