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A wireless mesh network (WMN) is expected to be a key enabler for next generation networking due to its infrastructure-less strengths such as scalability, costefficient rapid deployment, and long distance communications. To boost the strength of WMNs, it is required to balance loads among multiple gateways as well as mesh nodes. However, there have been only few practical and easily-implementable solutions for load balancing of WMNs. In this paper, we propose a distributed autonomous load balancing routing mechanism for anycast WMNs where most traffic flows are for the Internet access through any mesh gateways. Without flooding of control messages, our protocol finds optimized paths implicitly considering geographical distance and a congestion degree based on analogy of physics theory, Poisson's equation and by use of a distributed form of a finite element method. Through simulations, we assess the performance of our protocol compared with the anycast modifications of traditional shortest-path routing and geographic routing under dynamically varying network scenarios.